A lot of fanfic writers start up because of their passion for a show. While any number of things can be the source of passion, a very common fixation is on one of the primary characters. So they start to write something because of their passion for this character, but they find something about the character that doesn't mesh well. Maybe they're the wrong gender or are otherwise not close enough to the author's expectations. Or, alternatively, the author wants to pair their favorite character up with somebody, but "nobody is good enough for them". In any case, rather than put them through the Possession Sue process, they just get a Clone-O-Matic™ and out pops a Copy Cat Sue.
The Copy Cat Sue is a distressingly common occurrence, marked by their uncanny resemblance to a canon character. This is usually most blatant when it involves traits that are supposed to be unique in The Verse, but pretty much anything where the Backstory and key character traits of the two characters are nearly interchangable counts. The canon character is the last of their kind? Well, so is this character! Abusive Parents? This character can sympathize! The canon character has a unique Transformation Trinket? It's not unique any more! They might even have another Team Pet of the exact same species, and they get along with the canon pet absolutely fine! Naturally, they are instantly accepted into the group and have to go on a mission where they kill the Big Bad, slay The Dragon and, provided they're sexually compatible (not like it stops them from trying, though), they'll inevitably pair off while flipping the bird to the Official Couple. Or die. That happens sometimes.
A fairly common storyline that lampshades this is where the Big Bad (or, perhaps, just a neutral Mad Scientist; it's almost unheard of for anybody without a direct connection to the hero and his True Companions) makes an actual clone of the character (usually of the opposite sex, but not always). They play themselves off as family members from out of state, or exchange students from Mary Suetopia, but they have a mission to off the main character when they get the say so. Of course, since 99% of the Copy Cat Sues out there are synthesized to allow the author to imagine being both the canon character and friends/lovers with them, they'll betray their creator and join with the heroes (who, despite the character's connection with the bad guys, will rarely question them for very long), usually proceeding to kill the Big Bad, slay The Dragon, and become best friends/lovers until the end of time. Or die. That happens... a lot really.
A telltale sign of such a character is that any "fan art" of the character will actually just be recolored images of characters from the actual show and that may or may not explain why, say, Sailor Earth has a crescent moon on her forehead. Some get more creative and edit their source image a little in an attempt to make them look a little less like the character they copied. A precious few can be bothered to actually create real art, but nine times out of ten, even that art will simply be traced from a picture of the source character.
In either case, the character might be intended as a replacement for the canon character, but without whatever icky traits the author hates. They'll then rob the spotlight, prove the canon character to be unworthy of his/her position, and either relegate the character to obsolescence or, perhaps, even remove them entirely. Sometimes, this is done because the author likes the villain and wants them to be friends/lovers with the other character, but can't find any real reason that they wouldn't kill each other. Thus, to get the ending they want, they just make somebody that practically is the other character but without the key trait of wanting the villain dead. Yay, Fanfiction!
Compare Overused Copycat Character and Fountain of Expies.
The Bleach fandom have a few characters that end up getting their powers the exact same way Kurosaki Ichigo does going through the pit and everything in the exact sequence and having the exact same reason for going in.
Also, when some people try pairing Hitsugaya Toshiro with someone, due to his relative age, they make yet another child genius who goes to the academy at the same time as him, despite the fact that they are supposed to be rare.
The second movie, The Diamond Dust Rebellion, plays with this idea somewhat. In the film's backstory, Hitsugaya is shown to have had to a childhood friend from his academy days named Sojiro Kusaka. The two were in the same class, were friendly rivals when it came to school work, and even had the same Zanpakuto spirit (Hyorinmaru) as their partner, which only caused their friendship to grow deeper. However, things go south quickly when the Central 46 rules that no two people can possess the same Zanpakuto spirit, ordering a Duel to the Death between Hitsugaya and Kusaka. Kusaka charges at Hitsugaya, claiming to want Hyorinmaru's power for himself, revealing that the duel was a Secret Test of Character meant to determine who would raise their blade first. Kusaka is executed almost immediately afterward. Cue Kusaka coming Back from the Dead and returning as a menace to the Soul Society as Hitsugaya's Rival Turned Evil years later.
Likewise, there are nearly countlessDeath Note fanfics in which the main protagonist (or antagonist) is a genius detective with a single letter for an alias and who happen to be an orphan from Wammy's. Bonus points if they turn out to be L's, Near's, or Mello's long lost relative or their love interest.
Hey check out my Shinigami! It's liek big and totally awesome and eats fruit! Oh and we're liek best pals and it will gladly die for me if I asked!
Bonus points if the surviving Saiyan is a long lost member of the royal family.
Ironically, years later, a (anime original) canon royal really did come out of nowhere. However, it was a scrawny male who King Vegeta sent away for being weak and Vegeta "Jr" considered an embarrassment. He was also a pretty nice guy, and Happily Married to a Humanoid Alien (which, for some reason, seriously weirded Vegeta out).
Many fanfics in the Fullmetal Alchemist series have Ed and Al finding a lost long sister or childhood friend who they somehow forgot about. She'll often be a State Alchemist with some state name that makes no sense, be able to perform alchemy without transmutation circles, and have an automail limb just like Ed.
There are also tons of FMA fanfics that involve a new Homunculus character named "Jealousy" or "Desire" or "Anger". Bonus points if they're a love interest of another Homunculus and/or if they happen to have some sort of connection with the main cast.
Considering how many chimera fancharacters that show up who just happen to be working for Greed, it's a miracle the Devil's Nest isn't over-crowded by now.
Hamtaro falls to this as well. Most seem to be colorful versions of Hamtaro, Oxnard, and Bijou themselves. Most are paired up with the Ham-Hams as well. They can have a variety of other sue traits as well varying from Purity Sue to God-Mode Sue.
There are Hellsing fancharacters out there who are plucky female vampires that were bitten by Alucard himself and work for the Hellsing Organization and... yeah, they're pretty much Copy Cat Sue versions of Seras Victoria.
Because Tsuna of Katekyo Hitman Reborn! was initially disliked for being a Butt Monkey, several stories were written with him starting out as a genius instead of an idiot, and other things that replace his imperfections. Some authors go to the point of turning Tsuna into a girl for the sake of pairing him with the rest of the characters, or, if it is a crossover, make him know certain abilities that exist from other places beforehand, which would effectively make him a complete Story Breaker Power for most of the story.
Ouran High School Host Club OCs have a habit of being girls who, for whatever contrived reason, pose as boys and join the Host Club just like Haruhi. For bonus points, they may even be a missing Hitachiin sibling who acts remarkably like the twins.
The Pokémon fandom. It'd be fun to count, sometime, the number of trainers that start out late to get their Pokémon and get a non-standard starter. They'll likely be from Pallet Town as well, or even related to Ash with the writer's pairing of choice being the mother. Ash will, naturally, be a Pokémon master by then.
You could say this has to do with the games. If most of the people who wrote these fanfics even played the games, or if the characters were even any of the protagonists. Though whenever Brendan appears in an anime fanfic this happens.
Anime!Brendan is presumably Professor Birch's son, and he has a fully evolved water-type starter, which makes him remarkably similar to GARY MOTHERFUCKING OAK. He can easily be an OC Stand In for the rival when the fic takes place during the original series (Harrison from the Johto League would represent the player character.)
The first season of ''Pokémon also had two other trainers who left Pallet Town the same time as Gary and Ash. Similar to Brendan, they're OC Stand In fodder. Also subverted at the same time, as during the series it's said that they gave up and went home.
You cannot count the number of fanfics that have Ryoma'sLong Lost Sibling (always female), who looks exactly like him, plays the same tennis (better than him and his father even) and joins the BOYS tennis team because she is just that "awesome". Granted he has an adopted brother he barely remembered until recently but that doesn't give an excuse for the Echizen's sister fics running around.
Younger Sailor Earth creators in the Sailor Moon fandom have a tendency to copy a lot of traits from main character Usagi's appearance and backstory, sometimes, in worst cases, right down to being Mamoru's past-life lover. More often than not, said characters will also be Usagi's long-lost sister or cousin. More often, if the author does not want to bang Mamoru, Sailor Earth is Mamoru's sister from the Silver Millennium.
The YuYu Hakusho fandom... dear Kami-sama, the YuYu Hakusho fandom. Most of the fics that aren't pairing Kurama with Hiei, have: Kurama paired with a plant-weilder, kitsune, demon trapped in a human body after "death", thief, past mate and/or partner of Youko Kurama's, or any combination thereof.
Hiei is often paired with fire- or ice-wielders, hybrids of some kind (usually "forbidden" or "taboo"), working for Koenma/stuck in the human world, or possibly had a very rough childhood that made them very cynical and aggressive (any combination also applies). Not to mention that they are almost guaranteed to be at least a couple of inches shorter than him (no easy task, considering he is 4 foot 10 without the hair and they would have to be a legal midget).
There are also numerous delinquents from broken homes who barely go to school, talk back to authority, start random fights to prove how tough they are (especially when someone accuses them of being feminine), and somehow end up working for Koenma. Surprisingly, these are not often paired with Yusuke.
The Flash comics feature a canon example of a Copy Cat Sue (sort of). Barry Allen seemingly returned from the dead in 1993, but it turned out it was Eobard Thawne before he became Professor Zoom. Eobard Thawne was such a fan of Barry Allen, he memorized Barry's biography, spent years and his fortune to aquire the same powers, and had himself surgically altered to look exactly like Barry (using futuristic plastic surgery.) But he ended up arriving years after Barry's death and, addled from time travel, and the revelation that he would become Professor Zoom, snapped and started thinking he was Barry Allen. One can only imagine how well Barry would have reacted to the arrival of an obsessive Fan Boy who looks and dresses exactly like him and has his powers. If only DC did actual "What ifs".
Dozens of these popped up for Rise Of The Guardians within days of the movie release. The most common ones copy (and pair with) Jack Frost, all being teenage girls who die and become elemental spirits and new guardians; though there's also a fair number who copy Jamie, the first human to believe in and see Jack. Only the Sue is older, of course.
Films — Live Action
Lately on deviantART, there's been a massive Na'vi invasion going on. Doesn't really help that they all look nearly the same on first glance, and that tribal outfits are similar...
Star Wars fanfics often give Han Solo a sister (or brother?) who falls in love with Luke. A variant is to have Obi-Wan take a padawan on Tatooine, who then falls in love with Luke. These characters are very much Copycat Sues, usually a Distaff Counterpart of their relative, or a clone if male.
Minerva from Artemis Fowl is bashed for this. She's not too bad a character and Artemis beats her almost easily but she was too similar to the eponymous character and the author was trying to ship the two a little too much.
For what it's worth, that book is a Dolled-Up Installment. Colfer realized that his new protagonist was a bit like Artemis and just decided to throw him in, demoting Minerva to a secondary character.
In Harry Potter fanfiction, Harry often gets a twin brother who is in fact the "Boy Who Lived".
Gryffindor has won every single match since 1997 because it has 20,000,000 Seekers on its team.
There are several Harry Potter fanfics in which a female character becomes a student at Hogwarts. Said student is a female, happens to be really smart, (often) has Muggle parents, and...Yeah, it's a copy of Hermione.
Considering how many copy-cat sues of Draco Malfoy there are, it's a wonder Slytherin isn't over-crowded. Bonus points if said Slytherin character is Draco's relative or his love interest.
There are also tons of Weasley fancharacters as well. Yes, the Weasleys were a large family, but the number of copy-cat sues would be enough to populate a small island.
At least half of any prequel fanfics will contain some "lost" member of the Mauraders. Bonus points if they are a relative of a main character, or a love interest. Also, said fancharacters will have an Animagus form that's supiciously similar to the Animagus of a canon character.
Fancharacters that are female versions of the Mauraders are also prominent in fanfiction.
Oh, look, there's a new Defense Against The Dark Arks teacher who happens to be a werewolf. Many fanfics take this further by making said fancharacter a female and become Lupin's love interest.
A lot of protagonists from the Star Trek franchise's Expanded Universe and Fan Fic titles have beaten the designed-to-be-unbeatable Kobayashi Maru test, something in canon originally only Captain James T. Kirk had managed, making this seemingly impossible and impressive feat in fact depressingly mundane.
Almost every Twilight OC is a perfectly normal girl who just happens to move to Forks, Washington and become a new student at Forks High School. Despite always describing herself as plain and perfectly normal, she somehow attracts the attention of the beautiful Edward Cullen, who is intrigued by the fact that he can't read her mind. A particularly notable example of this is the heroine of Forbiden Fruit The Tempation Of Edward Cullen, who, um, "temps" Edward away from Bella.
Live Action TV
Big Wolf on Campus. Oh, look, a perky goth character who just happens to be a female counterpart for Merton to fall in love with.
There sure are a lot of female werewolves running around Pleasantville that Tommy seems to always fall in love with. Good thing they're good werewolves like him too, especially since heroic werewolves are extremely rare in the canon of the series.
Due to his sheer popularity in the fandom, Mr.InVinceible has quite a few counterparts in fanfics.
A lot of Doctor Who fanfics involve The Doctor running into a surviving Time Lord who just happens to have a TARDIS as well and travels across time and space with a companion. Bonus points for if said Time Lord is either The Doctor's long-lost relative or becomes his love interest. When you consider that The Doctor essentially wiped-out his entire species (Save for the number of times The Master shows up), one has to wonder exactly where these other Time Lords are coming from.
There are quite a few fan-made enemies of The Doctor running around with "The (insert something here)" as their name. Bonus points if they're the love interest or a relative of The Master.
Played with on Glee with Harmony and the other applicants for the school Kurt and Rachel were trying to get into in the third season.
A parody example comes from FoxTrot: one series of strips had Jason suggest to Lucasfilm that they spice up the Star Wars Special Editions by adding a new character: Jason Skywalker, Luke's younger brother who follows Luke's character arc up to the end of Empire, at which point he joins the Dark Side (and of course, lives to fight another day).
Some people doing fan art from the Final Fantasy series draw their own flavors of the moogles, chocobos, or any other race/critters that pops up in the series. These aren't as bad as the Sonic or Spyro examples above since characters like the moogles are usually generic and implied to be more than one of them running around, so the characters the artists use for their own characters aren't unique or special.
Final Fantasy Tactics Advance intentionally invokes this trope with Llednar Twem, Mewt's totally badass Marty StuDragon/friend, who bears a suspicious obvious resemblance to Marche. While Marche is fighting to destroy the fantasy world Mewt created, Llednar is there to oppose him.
Kingdom Hearts has quite a number of Sues which claim to be Sora's sister, given a mysterious-but-very-Keyblade-like magical weapon (usually when Sora receives the real Keyblade) and journey along with their "brother." Usually, they get shipped with Riku. Then 358/2 Days came out, which introduced Xion, who fit quite a few of these traits — spitting image of Kairi, fourteenth member of Organization XIII, a Keyblade user, and ends up being close friends of Roxas and Axel to boot. However, this ends up being deconstructed when it is revealed that she is an imperfect Replica of Sora created by Organization XIII to absorb memories from Roxas and eventually absorb him completely in case the real Sora was useless to their plans. She knows none of this, but when she does find out, the results are...unpleasant as she frantically tries to fix the situation, while desperately trying to avoid hurting her friends in any way (this fails... miserably).
They seem to love to destroy this trope. They had Namine, who looks just like Kairi with blonde hair and was a friend of Sora who he forgot about. Later it turns out that none of that is true. She's a Nobody created when Kairi's heart left Sora's body and she has the power to change his memories due to her origins. The Organization then forces her to do so. They even go so far to try to completely replace Kairi with her.
Also in Chain of Memories there is Replica Riku who is brainwashed into beliving that he's the real Riku. When he finds out he's not too pleased.
Many a Pokémon game fanfic stars trainers from small towns who defeat evil teams, defeat everyone in sight, become the champion, beat the game protagonists, and typically have either a Starter or an Eevee. They're often also a love interest to a game character.
Pokémon goes through a similar example when it comes to the creatures themselves but plays out in two ways. The not so harmful way has fan artists draw their own flavor of Pikachu or any other Pokemon they like with their own quirks since it is established that Pokemon breed, thus there's always multiples of the same species running around. The other method, which has been compared to Sonic color swapping, is people making their own version of Mewtwo or any other unique Pokemon (unique=the only one of its kind).
Portal Fanfics are surprisingly bereft of this, despite the fact that copy catting in non sue forms is very prevalent (Most of the characters are AIs and for the rest Aperture just has cloning, so if you need an extra version of a character; they are just a button push away). However you can consider most if not all OC test subject sues to just be Copy Cat Chells.
Albert Wesker sure does have a lot of clones running around in the fandom, doesn't he?
To twist the knife further, Resident Evil 6 features Wesker's son Jake Muller as one of the main protagonists. And while he's not morally twisted like his father (just an Anti-Hero), he's definitely inherited Wesker's speed, strength, and durability.
Sonic the Hedgehog fandom is notorious for this. Recolours and sprite edits make up most of the fancharacters, and the availability of character creators and paper doll programs to facilitate this have worsened the issue. Seems to be a fully Discredited Trope at this point, not that it's gonna stop a lot of people.
Though Jazz Jackrabbit is not one of these, there is a boss in the first game that looks like a terribly recolored Sonic/Tails/Zool mix.
Generally speaking, it seems that anybody who makes a dragon character (especially female) uses Spyro & Cynder's design.
And if that wasn't enough, some of these characters break up the romance between characters. A lot. Did I also mention that most of the creators also pass them off as OriginalCharacters as well, in which they have their own universe and storyline, but still look an awful lot like a Spyro or Cynder recolor. They even use The Legend of in their titles to add to the confusion!
A few exist for the Star Fox series. A lot of them are VERY similar to the main hero, Fox McCloud, have a similar level of skill as a pilot, or as an added option can be Spear Counterparts to Krystal, who prove to be even better love interests for her than Fox is (because y'know, Fox totally sucks).
Noticable in Super Mario Bros., though mostly less frequent than other examples. There are three specific variants, two being more princesses and Paper Mario partners. Those are usually not too sue-ish, unlike the less common Mario Brother clones.
In Avatar The Last Airbender, many a fic revolves around the group finding an Airbender out in the middle of the woods, raised by themselves, abandoned by the world. They get free lessons from Aang, just because he cares. Lessons for other Avatars cost extra, presumably for breaking the one avatar a generation rule. It's worth a mention that there is a (presumably non-canon) character from the Collectible Card Game that is, in fact, an airbender who had survived the genocide. This character, known as Malu the Ghost Witch, was hidden by her parents inside a cave, and quickly became a legend amongst the Avatar world for attacking Fire Nation soldiers.
Many "Chipmunk Bands" on Youtube are like this. Basically they're just the Chipmunks and Chipettes with a different color scheme and (most of the time) some random word in place of "Chip" on the band name (ex.: the Popettes, the Crystamunks, etc.). Also, Alvin colorovers are usually given some random "A" name (like Andrew or AJ) and most of the Brittany colorovers around are friends. Also, they have the tendency to Jump the Shark in many ways, such as getting married, having kids, or even having a member nearly die (like with the Omegamunks (Tim Ronald Jake) and the Pinkettes). Nearly. Most of the time they survive.
Ben 10: Even though it is confirmed that the Omnitrix is the only one in existence, in Fan Fic, every other kid has one or a mystical transformation artifact shaped like one. Word Of God quickly jossed every single one of them in the episode Good Copy Bad Copy, going so far as to say that more than one Omnitrix at a time would be The End of the World as We Know It.
Albedo himself, who appeared in the episode, also appears to be a parody of the idea, who helped assist in creating the Omnitrix, came back with a better version, and his current design is just a mere recolor of Ben (amusingly, Albedo has since become quite popular amongst the fans).
Oh gosh, the Code Lyoko fanfiction. So many stories revolve around these things:
An Original Character (or the author) comes to Kadic, befriends the Lyoko Gang, and becomes a Lyoko warrior
An Original Character is found on Lyoko, and has similarities to some new student at the school.
A group of original characters, either the original gang's kids, or some new kids who discover the factory and some wise guy pulls the lever and discovers Lyoko.
The Warriors fighting a brand-new supervirus that acts like XANA, but clearly isn't, really.
Bonus points if any of these characters (including the supervirus) fall in love with a Lyoko Warrior. Further bonus points of it's Odd.
In fact, if a story's main plot doesn't fall into any of these categories, it usually falls into the remaining 10% of Sturgeon's Law by default.
Wilt: Some kids aren't that creative and just copy what they see on TV. What're you gonna do?
Gargoyles had this as the reaction to canon character Preston Vogel, who just seemed like a darker-haired version of OwenBurnett. The only difference in their designs was literally their hair color (and Owen later getting a stone fist). Turned out Owen or more accurately, his real self the fey Puck had based himself off of Vogel intentionally, trying to "out-Vogal" him for personal satisfaction.
Invader Zim's Tak doesn't seem to be the only Irken female to wind up disguised as a human in Ms. Bitters' class, at least if you go by fanfic portrayals. There are also a fair number of Distaff Counterparts for Dib, random new girls who quickly see through Zim's disguise and either help Dib fight him or help Zim change through The Power of Love.
The original series were just as bad, mainly My Little Pony N Friends and the G1 toys. It's hard to find any fanart or fanfiction pony related that wasn't based around a ponification or OC.
Interestingly, the background ponies in My Little Pony: Friendship is Magic have no canon personalities or even canon names (toy names aside), but the fans as a whole seem to agree on certain names and personalities.
With the introduction of Discord, there have been tons of Draconequis fancharacters that all seem to have a name that relates to "chaos" in some way (IE: Disorder, Eris, Anarchy, etc.). Bonus points if they end up being Discord's love interest.
Despite Word Of God stating that there are only two Alicorns (IE: Luna and Celestia) in Equestria, that hasn't stopped fans from creating their own Alicorn fancharacters. The introduction of a third alicorn, Cadance, in the Season 2 Finale probably did not help this trend.
Making Twilight an alicorn in the season 3 finale definitely didn't help this thread either.
Fan fictions of The Powerpuff Girls became littered with specially created Puffs (whose names begin with "B," of course). Lampshaded in comic book issue #50's Deja View (originally a season 5 episode that didn't get made) which features alternate universe versions of the girls with the names Berserk, Brat and Brute. Also with the Micro-Puffs (issue #65 and others), tiny versions of the girls from another dimension.
South Park fan characters were rather infamous and known for consisting of recolors or genderbends of the four main characters, if not having them be related to any of the four in some way as long-lost siblings or love interests. After a character generator came into existence, fans went even more wild with it. There tended to be quite a few copies of Kenny, all dressed in different colored parkas and with the same level of immortality and bad luck. Cartman would occasionally wind up with a Fat Girl copy of himself, who would be just as rude and foul-mouthed. Some authors would also follow a "next generation" formula and have the future children of the originals be nothing but clones in personality and appearance.
Sponge Bob Square Pants has a surprising amount of these. Some even claim to know everything about the Krusty Krab even if they've never been there before or have never even been in Bikini Bottom up until that point! Usually they're copies of either Spongebob, Patrick, Squidward or Sandy, but Plankton and even Gary lookalikes aren't unheard of. They all are either somehow related to them or get paired with them eventually.
Blueberry Muffin deliberates makes one in the Strawberry Shortcake's Berry Bitty Adventures episode "Partners in Crime":
Blueberry: So I had this idea for a brand new detective: Peggy Periwinkle! What do you think? Huck: Sounds great, but isn't it a little bit like Patty Persimmon? Blueberry: As close as I could get! Huck: But don't you think we should invent something, I don't know, different? Blueberry: Oh no, that's the whole idea! Because anyone who likes Patty Persimmon will automatically love Peggy Periwinkle.
Superjail! occasionally experiences these in both fanfic and fanart. Lord Stingray will suddenly gain a girlfriend or long-lost sister who wears identical gold armor but will wear a bow and pink or purple to denote her being more feminine. The Warden himself tends to accumulate mysterious offspring who are carbon-copy clones of him in appearance and personality, if not experiencing his own case of having a long-lost identical sister (when fans aren't putting Mistress in that role). Another common type is the "new girl" at the jail, who tend to be quirky and wild love interests for the Warden and somehow all wind up accidentally dragged into the jail by Jailbot fighting Jacknife in their usual escapades.
Before and after their revelation as aliens, some fans have the Twins wind up with a younger sister tagging along, or a special female member of their species playing the part of love interest (or sometimes both roles).
Teenage Mutant Ninja Turtles has this. If a Sue is in the fic, they'll probably be a turtle, and wield weapons identical to the ones their favorite character does, and probably have a similarly coloured bandana too.
An special example is the fanfic The Hunters and the Hunted - it has FOUR Copy Cat Sues, one for each Turtle. Better still, the author gave them bios in the first chapter so you don't have to read through the whole story to find out - although you do have to read through the whole story to find out just how bad the author is at explaining why they never use their vast repertoire of abilities except when it looks cool, and never to get them out of danger. Hello, Chickification.
It is common in Teen Titans fanfiction to have a Raven/Terra clone join the Titans for various reasons. Raven is usually bitchy to this newcomer, but slowly warms to her. Or Raven may be completely Out of Character and not do much of anything. She usually has the same powers as Raven, complete with the emotional tie-ins. The clone will usually fall in love with Beast Boy. Right after the season where Terra betrayed the Titans for Slade, it was especially common for said clone to be Slade's replacement for Terra.
Other times, the new Titan will be a never-before-seen sibling of one of the main five, usually having some corresponding name to the original and being nothing but an opposite-sex copy.
Thomas the Tank Engine fanfictions have a lot of this. There are a lot of Thomas, Emily, and Rosie lookalikes running around on the Island of Sodor, it seems...
If it's a Transformers Sue, there's a good chance that they share a vehicle mode with their romantic target of choice. Especially annoying given the potential for creativity in the premise...
Somewhat more excusable here than in other fandoms, due to Hasbro's hard-on for Palette Swapping that just won't go away.
David Gonterman's prized Johnathan Brisby is, perhaps, the most blatant example of this trope in action. In his original incarnation in The Rangers of NIMH (a Crossover fanfic of Chip 'n Dale Rescue Rangers and The Secret Of NIMH), he's essentially a Generation Xerox of the canon Jonathan Brisby (but spunkier!). Then he undergoes Canon Migration (so to speak) to become a shameless ripoff of the character in Fauna Force. This will no doubt be the case for whatever upcoming story Gonterman has planned for the re-envisioned "Johnny Briz".