Yugi, Bakura, and Marik in Yu-Gi-Oh!. Though not Millennium item-related, Kaiba and Set(o) might also count.
Dragon Ball Z fanfiction Honor Trip has this in the form of Cell (who is now a protagonist), & Future Cell, who is most decidedly NOT.
Jamie "Multiple Man" Madrox from the Marvel Universe falls into this on occasion. For instance, when he found out one of his dupes had lived a full life without him (even to the point of having a son), he was stunned that "I named you after my father..."
In an issue of X Factor, Jamie Madrox split up into opposing groups of Multiple Men. Upon encountering each other, one of the Multiple Men said, "There I am! Get me!"
In the case where a clone needs to be distinguished to an earlier version of himself / herself who is no longer alive, the common usage in Cyteen and Regenesis is to refer to the original as "my / your predecessor" if they have the same name. Another usage is to refer to the elder of the pair as "senior", as in "Warrick" versus "Warrick senior".
Azi - the mass-produced clones who are the most common result of the technology - grow up knowing that there may be many copies of their particular geneset / psychset combination; that is, many clones with an identical environment, leading to very, very similar people. In Regenesis, the current version of Florian has to discuss his predecessor. He reflects inwardly that time before he himself existed was not emotionally attractive to him; he has no trouble with the fact that they're really not the same individual, however similar they are.
Parental replicates - clones of people who had a "normal" upbringing as opposed to the structured education of an azi - are much harder to replicate psychologically, and do sometimes have trouble coping with who's who. In the case of a parental replicate, where the original and the replicate have the same name, the latter may be referred to with the suffix PR attached to his or her name.
Android at Arms (1971): The protagonists wake on a strange world and learn that they have been kidnapped and stored as Human Popsicles, while being replaced by android duplicates. The question eventually arises, which one is the Robot Me (a Ridiculously Human Robot by necessity) and which is the original, and how to prove it? When the main character, Andas, confronts a much older version of himself on his homeworld, both are deeply shaken - each believes he's real, but how could a Robot Me be such a Ridiculously Human Robot as to do the various things each has done? (The protagonist refers to his older counterpart as "the false Andas"). Another variation happens later, when he and one of his companions wind up in an Alternate Universe, and he confronts a dying version of himself.
Star Gate (1958): all the human colonists on Gorth evacuate the planet at the beginning of the book because The World Is Not Ready - that is, the native intelligent species of Gorth (who call them the Star Lords) isn't ready for the humans' much more advanced technology. Some opt to search for an Alternate Universe in which Gorth never developed intelligent life. They accidentally wind up in a Mirror Universe in which their counterparts enslaved the natives rather than helping them. The Half-Human Hybrid protagonist refers to the Mirror Universe counterparts of the Star Lords as the Dark Ones, the Dark Lords, or (in the case of individuals, such as Lord Dillan) "the false Lord X" or "the Dark Lord X" to distinguish them from the "true" Lord X. (The Lord of the Rings was less than 5 years old at this point, please note.) The eldest of the Star Lords has the hardest time adjusting to it when he finally sees the Dark Lords, because although he knew intellectually what they were, it hit him very hard to see (apparently) several people among them who in his universe were long dead and had meant a great deal to him. He had to be restrained from going to them until he got himself under control.
Victory on Janus (1966): Big Bad THAT WHICH ABIDES creates android duplicates of the Iftin and of some human colonists - not as Evil Knockoffs, but to frame the Iftin for apparently attacking the colonists. The Iftin refer to the android duplicates as the "false Iftin", and can tell them apart from the true ones by smell. The worst problem the protagonist has is when duplicates turn up, not of himself, but of an old Love Interest and an old friend, both probably, but not certainly, long dead.
Averted by the Fel clones in the Hand of Thrawn duology. They have major, majorclone angst, but each of them has named himself something different, dresses differently, and has different hair. They keep a common family name, Devist, and let people think of them as just a really close and fairly insular family, because they're desperately afraid of being discovered and killed for being clones. A brief stretch where one, Carib, is the viewpoint character and has a momentary existential meltdown ends as he strengthens his resolve by reminding himself what he's worked for.
In The Time Traveler's Wife, reluctant time-traveler Henry keeps running into versions of himself in the past, present, and future. Can be shocking, funny, sexy, or tragic, depending entirely on the circumstances.
In Brothers in Arms, Miles Vorkosigan discovers he has a clone, who has been trained to impersonate him and is also (at this time, anyway) known as "Miles". At one point, the two clones are chasing around a large building, leading Miles to ask a bystander "Have you seen me already? Which way did I go?"
Intrigues are all about it: Mikuru goes one week back in time and poses as her own long-lost twin sister.
And in the climax of Astonishment when two versions of Kyon from two parallel realities meet in a closed space.
In The 7th Voyage of Ijon Tichy a.k.a. 147 Vortexes by Stanislaw Lem the titular space traveler cannot fix his ship alone and drives it into a gravity vortex to create a time loop that would bring another copy of him. Then into another to copy a spacesuit too. Unfortunately, the copies cannot agree on anything, keep beating each other, stealing chocolate and lumping the blame on "the Sunday one". And the poorly controlled ship keeps running into more vortexes bringing more Ijons from different times, including an old man, who tries to be an arbiter, but turns out too senile, and 2 teenagers who end up saving the day.
Common in the works of authors like Linda Nagata, Charles Stross, and Greg Egan where there's no technical reason that software intelligences can't duplicate as many copies of themselves as needed. Stories set when the technology is new or in settings where biological and software intelligences interact lead to confusion.
In Nagata's Vast Clementine is dismayed by the way Nikko and Urban casually create and destroy instances of themselves.
In Stross's Accelerando there are numerous personality clashes between people who treat persinality as fungible and those who cling to a single persona.
In Egan's Permutation City Peer spawns a clone of himself to enter the construct, and is faced with the decision of who will remain in the real world.
And in the fourth season, Hiro goes back in time to the period of the first, talks to his younger self, then calls him(self) a moron after sending him(self) off on a mission intended to avert paradox.
In an episode of Quantum Leap, Sam leaps into a young Al, and they have to distinguish between observer Al and young Al. After a short while, Sam just suggests calling the young Al "Bingo", his nick-name at the time.
In the Red Dwarf episode "Stasis Leak", multiple copies of the crew begin to accumulate in the same time period, leading to general confusion.
Past Rimmer:Three Listers! Splendid! Perhaps Lister here would like to go over to the fridge and open a bottle of wine for Lister and Lister! Rimmer here doesn't drink, because he's dead, but Iwouldn't mind a glass!
Occurs multiple times in Stargate SG-1, at one time with the entire SG-1 team.
Happens in Doctor Who more often than not, with all the time travel, alternate dimensions and clones the show's got going on. It's also meant that fans have had to come up with ways of keeping things straight. The Doctor's multiple bodies are referred to by numbered regenerations- for example, Tom Baker is 'Four' and Matt Smith is 'Eleven'. When they're the same regeneration, other identifying markers are used- for example, the two versions of the Doctor running around in Season Five are called 'Normal Doctor' and 'Jacket!Doctor'. The Master is recognized by the actor playing him, and Romana is kept straight by numbering her as Romana-1 and Romana-2. River Song's bodies are kept straight as 'Melody Pond', 'Mels' and 'River Song'.
The two parter "The Rebel Flesh" and "The Almost People" brings us identical clones of various characters. These are referred to as Ganger!*insert-character-here*.
The clone of the Doctor made in "Journey's End" is usually called either 10.5, TenToo, or Handy The Wonder Clone. To tell them apart, the normal Doctor wears a brown suit, while 10.5 is in a blue suit.
In both RaidouKuzunoha games, an alternate version of the titular character appears. It's somewhat mitigated by the fact said alternate does not have the exact same name, but it still causes a great deal of confusion.
In Super Mario Galaxy, if you're playing as Mario, you find Luigi in various levels who refers to Mario as "bro". If you're playing as Luigi... you find Luigi in the same places, who refers to (player) Luigi as "me" in the same places in the dialogue.
The wormgate system creates a perfect duplicate of anyone sent through it, which is kept for interrogation by the Gatekeepers, who then kill the clone. The first characters who suffer from this problem are Doythaban and his gateclone Haban II, but this later becomes a galaxy-wide problem when billions of these clones are released. However, no-one suffers from it more than Gav, who clones himself 950 million times to escape, leading to an truly epic case of this trope.
Gav: There are still over nine hundred million Gav clones out there. My activities of the last year can only be understood statistically.
At one point the Terran government tries to charge Kevyn with treason for mass-releasing the teraport designs. He points out that it was his now-deceased clone who released that information, not him, and thus he can't be charged with anything.
It also leads to some rather bizarre court cases since there are some legal issues where the gate clones are not always considered separate individuals. In one case, a person had two death penalties against him for Manual Operation under the Influence. When his gate clone turns up, the judge rules that, since the clone was created after the commission of the crime, it is perfectly legal for them to apply the second death penalty to the gate clone.
Judge: Oh, and you used up all your appeals the first time around. Sorry.
In the "A Hand Full of Aceys" storyline, a gate clone attempts to murder the original version of himself, but instead gets killed by the original acting in self defense. Since the clone and the original are legally the same person in that particular jurisdiction, the final police verdict ends up being attempted suicide.
The Gavs eventually found a way to differentiate themselves from one another, to the point where they are barely recognizable as Gav-clones. Especially the females.
Pretty thoroughly averted in The Inexplicable Adventures of Bob!. Golly and Jolly are both biological clones of Molly with totally different upbringings, and the castmembers are apparently all bright enough to realize that even if the three look alike, they are different people.
In Bob and George, due to convoluted Time Travel and Alternate Universes, one story arc involves Mega Man and Bass, Past Mega Man, Alternate Mega Man and Bass, Future Alternate Mega Man and Bass, and Far Future Alternate Mega Man and Bass (though the last one was a trick). They end up wearing helpful labels to show who is who, since they all look identical in a Sprite Comic.
Every player has a dream self on the moon of Derse or Prospit respectively. Then time shenanigans get involved, both via literal time travel and time traveling messages. Then some of the dead selves end up coming back to life as separate characters from the "main" characters. It only gets more confusing when alternate universe counterparts to the heroes show up.
Then there's Pickle Inspector in Problem Sleuth. Thigs start getting out of hand with Past Future Pickle Inspector, and in the end an infinite number of his past and future selves become the whole universe.
Cerene runs into this problem constantly, because there are three of her. And yes, it confuses all three of them. (Or at least the two we've seen so far)
Los Hermanos of the Global Guardians PBEM Universe has this happen all the time. Since all his duplicates are him (they are all part of a mass-mind effect, and what one duplicate knows, they all know) he has no trouble keeping track, but his companions sometimes fall directly into this trope.
Bungie: "Bad news... um... you've been captured. I saw a couple of Tarot goons dragging... um... you... away toward their APC."
Los Hermanos: "Yes, I know... they've got a bag over my head, so I can't see where they are taking me. I'm a little roughed up, so not seriously hurt. They're probably taking me somewhere to torture me."
Bungie: (beat) "Right... sorry..."
Jade Sinclair (Generator) and her creations in the Whateley Universe. Jade's creations aren't really inventions. They're objects populated by a psychic clone of herself, including the character Shroud, who is claiming to be her sister Jinn. Even Jade can't keep her pronouns straight when talking about her selves, to the point that it is a Running Gag.
College!Phoenix, Hobo!Phoenix, and Lawyer!Phoenix in Phoenix Wright Ace Idiot. "I traveled back in time to murder my younger self, blamed it on myself, and hired myself to defend myself. Then I prosecuted myself to insure myself the murder of myself would never be linked to myself."
Happens in Ben10 episode "Ben 10,000" when Ben meets his older self.
Happens at the end of the Time Travel episode of Cow and Chicken, with Chicken complaining of his time-travel duplicate "I ate all the cereal, now there's none left for me!"
In The Curse of the Were-Rabbit, Wallace starts referring to Hutch in the first person after he takes on his personality. For instance, when Lady Tottington rings the doorbell, we get this:
Wallace (transforming): I can't answer the door like this!
Hutch (walking towards door): Charmed. I'm Wallace.
Wallace (yelps): I already am!
The Firesign Theater's audio play "The Further Adventures of Nick Danger" features a flashback told by Catherwood the butler - twenty years earlier he surprises his bride Nancy with a time machine, intending to honeymoon in ancient Greece. He tries it on himself first and when he returns minutes later he's a shaky-voiced 1,000 year old codger, to her horror. Present-day Nick bursts in unexpectedly and uses the time machine to travel forward 20 years back to the present, taking the couple and shady character Rocky Rococo with him. Back in the present they're shocked at finding there's two of all of them - they fight with their other selves, except the two Catherwoods who enjoy having someone their own age to talk with.
A fairly well-known T-shirt design reads: "I have gone to find myself. If I get back before I return, keep me here."