Uncommonly Common Exotic Pet: Seen It a Million Times: it is very common for fictional characters (especially if they are a bit Sue-ish) to own pets that in Real Life, are not common at all. Likely because, due to the Rule Of Cool, most of us have daydreamed of owning such a pet at one point in our lives.
Alien Pet: Why own a critically endangered animal when you can own a creature that's not of this world?
Mythical Pet: a creature or character that is actually a powerful mythological creature but treated as a regular pet. Muggles might actually see an odd-looking peacock instead of a phoenix, a horse instead of a unicorn, etc.
Impossible Pet: due to sloppy research, a character owns a pet whose care would lead to Fridge Logic.
This can be a Truth in Television since some brave souls do keep rather... exotic... pets. Some foolish souls do that too, but without actually knowing what they're doing.
Can overlap with Action Pet.
Compare and Contrast Mons.
There's combos like Xanxus and Bester, his pet liger box weapon who also shows that it's a Bond Creature when it changes from a lion to said liger form when Xanxus' scars show; Gokudera and Uri, who are both loose cannons with a bit of Cats Are Mean; and even the Arcobaleno all seem to have exotic companions.
Puu from YuYu Hakusho eventually morphs into this for Yusuke, getting its final form: a blazing phoenix.
Nagi Sanzenin of Hayate the Combat Butler has a pet tiger, Tama, who she adopted as a cub. He's able to talk, but Nagi doesn't know that.
Kyou from Clannad keeps a wild boar known as Botan. For most of the story he's just a piglet, but even so, he's extremely well-behaved and trained (excepting his habit of following Kyou to school), especially for a boar that was born wild, not in captivity. In After Story (by which point he's full-grown, tusked, and fairly huge) he's even seen playing with five-year-old children!
Runaways has Old Lace, a genetically engineered dinosaur from the future.
Lockjaw, the giant, teleporting bulldog belonging to Black Bolt of The Inhumans.
In Watchmen, Ozymandias has a genetically-engineered lynx named Bubastis.
Superman has Krypto the Superdog, a little white dog who happens to be one of the most powerful living creatures on Earth, perhaps in the whole universe. And he wears a cute little red cape.
Other DC heroes also had cool pets back in the day. Batman had Ace the Bat-Hound, Supergirl had Streaky the Supercat as well as a superhorse named Comet, and Superman also had a supermonkey named Beppo. The Kryptonian pets and several others also formed the Legion of Super-Pets in the Legion Of Super-Heroes 30th century.
Most of the mutants in Teenage Mutant Ninja Turtles were kept as cool pets at one point or another. Leatherhead was an exotic pet who was flushed down the toilet when his owners couldn't/wouldn't keep him anymore, there are the titular turtles (of course), and Master Splinter who qualifies not by being a rat, but by being a rat who knows martial arts.
Baron Winters, leader of Night Force, has a pet leopard named Merlin.
Red Planet has Willis, who is the low intelligence (compared to most people but more intelligent than a dog) pet of the protagonist Jim Marlowe. Later we find out that Willis is a female Martian who will grow up to be male; and will become equal to or greater in intelligence than humans.
The Star Beast's titular character is Lummox, who has been a pet to generations of the Thomas family. She's actually an alien princess whose species is noted for having difficulty picking up languages other than their own, and who views the boys of the Thomas family as her pets.
In Stirling Lanier's Hiero's Journey, Hiero had Klootz, a giant mutated, combat-trained riding-moose. Not a pet, exactly, but not quite a partner. And seriously cool.
The Talents series has Barque Cats (semi-sentient space-born felines) and Coonies (raccoons genetically modified for increased intelligence and domestication.)
In Andre Norton's The Zero Stone and sequel Uncharted Stars protagonist Murdoc has Eet, improbable telepathic offspring of feline parthenogenesis. She's basically a funny-looking kitten...but at the end of the second book, she metamorphoses.
Pretty much required in Philip Pullman's His Dark Materials, where the daemon is a physical manifestaion of your soul and generally equals your badassness capacity.
Mister, Harry's adopted tailless house cat that's the next best thing to an average mountain lion. It's an in-Dresdenverse Running Gag that Mister "owns" the house and that Harry just lives there. Bob occasionally uses him as a mobile Spirit Jar to protect himself from sunlight. Harry speculates that Mister only allows it because he gets to see interesting places.
In AnimorphsSixth Ranger David has a pet cobra named Spawn and a cat that isn't afraid to throw down with eagles named Megadeth. Jake has a less exotic but no less cool pet - a golden retriever named Homer.
Harry Potter had Hedwig, a snowy owl, which while not as unusual as some of the examples here, still qualifies because Hedwig was far more social and generally better-behaved than any real-life pet owl could be. Or other owls in the Potterverse, such as Ron's Pigwidgeon.
Also Dumbledore's Phoenix, Fawkes, and Voldemort's snake Nagini. Oh, and Hagrid's hippogriff Buckbeak, his threstrals, and his spider Aragog.
Hagrid also has Norbert the Norwegian Ridgeback for a short time.
Skeeve from the Myth Adventures series acquires a pet dragon, Gleep, at the beginning of the series. Much later it's revealed that it was, in fact, the other way around.
In the Discworld series, swamp dragons are sometimes kept as pets by the Aristocratic and wealthy. They are notoriously difficult to care for, having extremely stringent dietary and environmental restrictions, and are prone to exploding. Later books introduce Lady Sybil Ramkin, who runs a rescue shelter for unwanted swamp dragons. They're effectively a parody of pure-bred horse and dog keeping by the upper-classes; as well as exotic pet keeping in general.
In A Wizard of Earthsea, Ged has an otak, a small, very shy wild creature that rides around in his hood and will tolerate almost no one else. When he's attacked at one point, it tries to protect him, screaming (this is notable because otaks have no voices). Ged is heartbroken when it dies.
At the end of A Game of Thrones, Dany gets three baby dragons. A part of her plotline involves trying to find two more riders for them when they're grown. Also, dragons are the sigil of House Targaryen.
The Stark children andJon all have enormous direwolves, which are more than pets or friends - they are a part of them. Direwolves, the sigil of House Stark, haven't been seen in living memory. Sansa loses hers early on, unfortunately, and Arya is separated from hers.
Skinchangers, like the Stark children, always have a cool pet of some kind to warg into. Varamyr has three wolves, a bear, and a shadowcat (the impracticality of owning the latter two is Lampshaded and Deconstructed). Orell has an eagle. Borroq has a large boar.
Doc Savage's sidekicks Monk and Ham each had one of these: Monk had a pig named Habeas Corpus, and Ham had an ape named Chemistry.
A more mundane example than most, but Louis Kehlweiler (one of Fred Vargas' main characters) has a pet toad called Bufo.
Ord in Sister Alice has a dozen intelligent, tame bears as pets, who have limited speech capacity. The Chamberlain estate previously had goat-like animals, which clung to sheer walls, though they are no longer around because the children kept temporarily dying from falling to their death while trying to ride them.
The short story Snow White Pony by Ardath Mayhar has a little girl who gets a white pony from her parents. It's really a unicorn who's been sent to restore to her the will to live.
In Doctor Who, the Doctor has K-9, a charming little low-budget robot dog who calls him "Master" in a sweet little electronic voice and shoots lasers out of his nose. The Doctor had two versions of K-9. K-9 Mark III appeared in the short lived spinoff K-9 and Company with Doctor Who companion Sarah Jane Smith, as well as in the Doctor Who stories "The Five Doctors" and "School Reunion". Following a Heroic Sacrifice, Mark III was replaced by Mark IV, who subsequently made several appearances on The Sarah Jane Adventures as Sarah Jane's, and later her son Luke's, pet.
Drizzt has Guenhwyvar, who is animal companion and nigh-immortal figurine-summoned monster at the same time.
Miliana da Sumbria got some beautiful "silly bird". Let's just say there was a good reason why it was called "sacred, untouchable, and extremely dangerous; avoid at all cost" in a treatise on all sorts of flying creatures including literally hellish ones.
Pokémon is the king of this trope. Seriously, in recent games (IE: Diamond/Pearl/Platinum) you are given the chance to capture Arceus (the Pokémon counterpart of God). Not only that, but in HeartGold and SoulSilver, the remakes of Gold and Silver, you can get that God to follow you around. Or to create for you a being who controls time, space or antimatter. "You" being a mere human.
Even better in the fifth gen where you can catch among other things; an Archeopteryx, a Hydra, what is basically the Iron Giant and one that is basically Mothra
King of Fighters' Rugal has a pet panther. His children, Adel and Rose, have a panther kitten (apparently the offspring of their father's pet).
Medabots is also relevant, and is pretty much the Mecha version of Pokémon.
Digimon is also relevant, and the PS1 game is made of awesome, although it has a shit ton of bugs and glitches.
Also, who could forget the little handheld consoles that kicked Tamigochis out the fucking window?
Rush, Treble, Tango, and Beat are robot animals from Mega Man. The first two can even fuse with Mega Man and Bass respectively to make them more powerful.
Rider in Fate/stay night is something of a nature/divine spirit rather than a Heroic one, and associates herself with beasts. Therefore, the 'Rider' part of her name comes in with her summoned pet, a Pegasus. Not the Pegasus, but it's still rather badass even if she only uses it once. It also ties into her true identity, but is obscure enough to not reveal it.
Final Fantasy VI has Shadow, with his dog Interceptor. He's a completely normal guard dog...who can block any physical attack and take no damage and whose own attacks ignore any defense.
Final Fantasy VIII has Rinoa's dog Angelo, who is used in her Limit Breaks and does things no normal dog is capable of.
Eiko carries around a moogle which, as it turns out, is actually a God, and goes completely batshit when two jesters try to engage in sexual activity with the young girl.
Kuja uses a silver dragon to fly from point A to B before stealing Cid's prototype airship. The dragon turns out to be Garland's pet, but he's never seen riding it, sadly—probably because he already has a Cool Airship of his own.
The manager of the Treno weapons shop in tries to be like Jabba the Hutt, in that he keeps a massive creature all cooped up under the shop, just to see people try to fight it.
In The Darkness, Jackie can summon little gremlins to do his bidding, who play about once their job is done.
In Manhunt, The Director keeps a psychotic, naked fat man with a pig's mask as a captive in a mansion attic, this is relevant to the trope, I don't care what you say.
In Metal Gear Solid 3, The End has a parrot which is actually catchable, killable, and edible. Upsetting the old bastard with the sight of a skeletal parrot makes him even harder to kill.
Reading the above made me think of Koromaru, a dog/party member who has a similar fighting style.
The hunter class in World of Warcraft is built upon this trope - many people roll hunters just for the pets! Everything from owls to tigers to bats to dinosaurs to dragonhawks can accompany a hunter from level 10 (1 from Cataclysm) to the end.
Other classes with pets are Warlocks, who employ demonic minions. Unholy Death Knights have pet zombies. Enhancement Shamans get Feral Spirits, two ghost wolves that can be summoned temporarily to help and heal the player character.
World of Warcraft also has companion pets that any class can own. However unlike hunter pets they don't fight (a few fights critters or certain other companions) and are really just for show. They can be anything from common pets like cats, parrots, rats and turtles, to wolves, baby dragons or even miniature versions of raid bosses.
Even among hunters, some pets are considered cooler than others, and the Beast Mastery talent specialization tree ends with the ability to tame 'exotic' beasts as well as the usual selection. Of particular note are the extremely rare Spirit Beasts, which are a highly prized exotic pet because of the difficulty in merely finding them.
A Boy and His Blob. Literally a big white blob which shape-shifts when you feed it differently flavored jelly beans.
In Triquetra Cats, Rain has Pito-Pita Chan, a Gen 5 Splio Beast, although in this reality it's not considered overly exotic, just uncommon (more akin to having a rabbit for a pet rather than a dog or a cat),
Sluggy Freelance Bun-bun was intended to be this, but turned out to be killer enough rabbit to avert Bears Are Bad News. Kiki the ferret is still this. Aylee has been this more often than not, but now that she's taken humanoid form, the ZHOAS takes on the role, at least in Kiki's interactions with it.
Virus of Exterminatus Now has a Chao named Blasphemy who absorbed demon DNA and is apparently smart enough to play Halo. It was also mentioned that he has an arco-flagellant combat cyborg named "Skippy" who's been in cryo since the time Virus forgot his deactivation code.
Davan from Something Positive owns Choochoo Bear, who is a 30-year old, boneless kitty and essentially a sapient version of The Blob , with a penchant for traveling drains in order to have various adventures.
Both Spike and Princess Celestia have a phoenix for a pet (Peewee and Philomena respectively), although Spike has since released Peewee into the wild to be raised by adult phoenixes.
A Running Gag on Animaniacs was "Wanna meet my pet?" When faced with some otherwise terrifying monster, Dot would ask the aforementioned question, holding a small box in her hands. From the box would emerge a disproportionately huge "pet" that would terrify the first one. On at least one occasion Dot's pet was a giant Jerry Lewis with an Alien-esque tongue (i.e. his tongue was a smaller Jerry Lewis) that said "Raar! Monster!"
Young Justice: Superboy not only gets a giant mechanical alien sphere that latches onto him like a puppy in "Bereft", but he also befriends a mutated white wolf in "Alpha Male" after removing the Explosive Leash electric collar that was forcing it and the other mutant animals to attack him and his teammates. Miss Martian attributes this to Superboy's nature as a "stray" like them.
Several of the princes and princess in Sofia The First have cool pets; Hildegard has a mink, Vivian has a dragon, Amber has a peacock, James has a baboon, and Xandir has an elephant.
As noted above, quite a number of people keep exotic pets in real life. Many of these acquire exotic pets primarily for the coolness factor; and are unprepared to deal with the difficult and often complex dietary and care requirements. Unlike fictional examples, this commonly results in poor health and drastically shortened lifespan; as well as animals being abandoned into the wild when their owners are unable or unwilling to continue to care for them. Many are also implicated in attacks on people, or spreading of non-native diseases and parasites to native animal populations. And finding veternarians able to treat exotic pets is extremely difficult, if not outright impossible. Other pets — such as rats, hamsters, ferrets, and mice — athough still technically classed as "exotic", are common enough and simple enough to care for that they present few if any challenges beyond those faced by dog and cat owners.
Exotic pets frequently undergo fads; where a pet which is featured on a television program or movie, or kept by a celebrity, triggers a flood of people obtaining similar pets. Recent fads have included pot-bellied pigs, sugar gliders, and pygmy hedgehogs. These usually die out quickly, and result in many abandoned or poorly treated animals. Occasionally, fad pets with less challenging requirements have caught on more widely. This usually results in better resources being developed, greatly decreasing the difficulty of care, and leading to even greater popularity. A classic example is the ferret, a rare exotic pet less than 2 decades ago; now increasingly commonplace as pre-made diet, housing, and other care supplies are more widely available.
Major advances have also been made in tropical fresh- and saltwater fish keeping; greatly increasing their popularity. As the technology continues to improve, even more exotic and "cool" creatures become available; including some that were effectively impossible to keep less than a decade previously.
Some of the most commonly abandoned exotic pets are Green Iguanas, Red-Eared Slider turtles, and Burmese Pythons; with the "cool" factor being the most common motivations for Burmese Pythons in particular. They are often advertised in pet shops as "simple to care for"; and both are commonly described as "even tempered" or having a "friendly personality". While the latter part may be true, the former definitely is not. While starting small, Green Iguanas get up to 6 feet long, and have highly specialized care requirements, including a very limited diet and the need for large amounts of UV light; and require far more attention than dogs and cats to become adequately socialized. Burmese pythons will reach between 15 and 20 feet as adults, weighing between 100 and 200 pounds; and go from eating mice and rats, to eating rabbits, chickens, and pigs.
Unlike dogs and cats, abandoned exotics rarely survive long in the wild, due to the dramatic differences in climate and food sources between their native environment, and where they're released. However, there are exceptions to this, and a few have been released in such numbers that they have managed to establish breeding populations and gone on to become major pests. Notable examples include goldfish and carp in many parts of the US and Australia; Indian Mynahs in parts of Australia; Burmese Pythons in Florida; and Red-Eared Slider turtles in the southern and western regions of the US. Many countries and US states ban particular exotic pets, in an attempt to prevent this problem.
Parrots are generally pretty cool pets. They're smart and social, you can train them to sit on your shoulder and do tricks, and they're pretty much the only animals that can talk. However, like other pets listed on the Real Life examples, many buyers just see the "cool" part of it without realizing what they're getting into. Parrots, especially the larger ones, are loud, demanding, and need constant attention. Caring for them is appropriately described as caring for a human toddler. And many live for several decades. Many parrots often get neglected and abandoned, often sent to zoos or animal shelters. Even smaller, easier species like parakeets need socialization to be good pets, though since they're so cheap (and thus "throwaway pets" to many) people are more willing to get rid of them.
However, on the bright side, there are people like Rexano who promote responsible ownership. Even they say how cool it is to have an exotic pet but state that not only do animals have their own temperaments and that owners have to go extra lengths to keep animals like a Caracal over just a standard cat like a Maine Coon. And that one has to socialize an animal (especially a skunk) a lot when it's young so that it will remain that way when it's older. Even Mountain Lions can be very friendly towards a responsible person they form a bond with. (It helps that Mountain Lions are more related to domestic housecats than actual lions.)
High on the current cool pet scale (at least in the southern US) are Sugar Gliders, which are basically (extremely high-maintenance) marsupial flying chipmunks
Paris Hilton owns a kinkajou, an exotic relative of the raccoon. It received media attention when it bit her, and criticisms over it also ensued over the fact that pet kinkajous are illegal in California, which did raise questions about how qualified she was to care for an animal with specialized care. Nonetheless, she was able to keep the kinkajou.