Delusions of Doghood

You see that guy there? He's crawling around on all fours, and...is he sniffing that guy's butt? Who does he think he is?

A dog, of course.

There's plenty of Shapeshifting tropes, but this one is a bit different. You see, these characters don't actually change form, they just think they do. This is whenever a character believes he is a dog or any other animal without actually being it.

Examples of Delusions of Doghood include:

Anime & Manga

The episode of Pokémon where a bunch of kids get accidentally hypnotized by a Hypno into thinking they're Pokémon.
- There's also one where Ash gets hypnotized into thinking he's a Bulbasaur.
Princess Mononoke's San, who adamantly believes she's a wolf and even wears a wolf pelt to look the part. Justified, since she was literally Raised by Wolves after her birth parents were caught intruding in Morro's forest and abandoned San, when she was an infant, to save themselves.

Comic Books

In New Mutants, Catseye of the Hellions thought she was an actual cat.
In X-Men, the Xavier Institute once had a student named Overlay, whose power was that he could induce this.

Film

Big part of The Animal.
The framing joke in Annie Hall is about someone like this.

I thought of that old joke, y'know, the, this... this guy goes to a psychiatrist and says, "Doc, uh, my brother's crazy; he thinks he's a chicken." And, uh, the doctor says, "Well, why don't you turn him in?" The guy says, "I would, but I need the eggs."

This is the overall theme of the early Nicolas Cage movie Birdy, in which Cage's friend believes he's a bird after nearly being killed at war.
The forgettable French comedy movie Didier is about a dog that's somehow turned into a human, but keeps a dog mind and Hilarity Ensues.
In High Anxiety, there's a mental patient who thinks he's a cocker spaniel.
In The Killer, Sidney's telling Jeff that he doesn't know whether he's a human or a dog anymore, which may a sign that he's close to (or has already crossed) the Despair Event Horizon as a result of the monstrously cruel No-Holds-Barred Beatdown he received at the hands of Johnny Weng, who repeatedly likened him [Sidney] to a dog during the beating.
Monty Python: the Silly Olympics includes the 3,000 Meter Steeplechase for People who Think They're Chickens.
This was a subplot in The Tracey Fragments. Like, she'd hypnotized her brother into thinking he was a dog, permanently or something. I'm too tired right now to double check that.
Wild Wild West. Artemus Gordon uses a spinning spiral disc to hyponotize General "Bloodbath" McGrath. He tells McGrath that he'll be Gordon's "little doggy" and commands him to speak. McGrath makes dog sounds, and Gordon tells him to speak in words. McGrath starts growling angrily and Gordon notices that his disc has stopped spinning. McGrath then attacks Gordon while growling and snarling at him.
The Disney film You Lucky Dog has a dog psychic played by Kirk Cameron who ends up inadvertently chanelling the personality of a rich man's St. Bernard whenever he gets excited.
In the sequel of the live-action One Hundred and One Dalmatians, there is a parrot who thinks he's a dog.

Literature

King Nebuchadnezzar from The Bible.
In Discworld books this is a speciality of witch Granny Weatherwax.

"Despite many threats, Granny Weatherwax had never turned anyone into a frog. The way she saw it, there was a technically less cruel but cheaper and much more satisfying thing you could do. You could leave them human and make them think they were a frog, which also provided much innocent entertainment for passers-by."

- Magrat:"I always felt sorry for Mr Wilkins...It was so sad watching him try to catch flies on his tongue."
- The Wee Free Men also has a frog who wonders if whether he's actually a man turned into a frog or a frog cursed into believing it. The former is true.

Live-Action TV

In one episode of 3rd Rock from the Sun Harry's pet from back on the homeworld tracks him down on Earth; the pet is transformed into a human (Bill Irwin) but acts like a dog.

Tommy: Oh, good. This'll help us be less conspicuous.

They cover by saying he took a bullet to the brain for Harry in "that war."

In Misfits, one person gains this as a superpower. They go through bouts of thinking they're a Jack Russel, but don't shapeshift.
Played with on How I Met Your Mother when the others point out that Robin's new date is basically a dog. While he doesn't act like a dog, his behavior is reminiscent of one (running after a squirrel, chasing his "tail", etc).

Newspaper Comics

In Calvin and Hobbes, Calvin can transmogrify himself into any kind of animal, but from his mom's perspective he's always shown as a six-year-old boy.
In Beetle Bailey, Beetle once hypnotises Sarge to think he's whatever he secretly wants to be, making him act like a lion. He also accidentally affects General Halftrack, but he doesn't "become" an animal: "Have you gone mad! I'm an airplane!" (Paraphrased.)

Webcomics

In this Order of the Stick comic, Haley uses a potion that increases her ability to bluff to the point that she can make anyone believe practically anything. Amongst other Blatant Lies, she deals with one guard by convincing him that he's a wallaby, causing him to irately hop off to find a wizard to undo the spell.
Inverted in Skin Horse with Phil the Mynah, an Uplifted Animal who thinks he's human.

Western Animation

In the Looney Tunes cartoon "Hare Brush", Elmer Fudd thinks he's a rabbit. He gets put in a mental institution, where he escapes, leaving Bugs Bunny in his place. Bugs gets convinced by the psychiatrists that he's "Elmer J. Fudd, millionaire. I own a mansion and a yacht." He then goes rabbit hunting, and Elmer is the rabbit he's hunting. Ultimately subverted, though, as Elmer!Bugs gets arrested for tax evasion. Elmer knows he's not a rabbit; he's faking it so he doesn't go to jail.

Elmer: I may be a scwewy wabbit, but I'm not going to Owcatwaz!

- In another of the Looney Tunes shorts starring Porky Pig and Charlie Dog, Porky is Driven to Madness and at the end starts acting like a dog.
In the Series Finale of Pound Puppies, the stress of running the pound (and inability to escape it, even when on an attempted vacation) eventually broke the evil aunt Katrina Stoneheart, causing her to conclude that maybe she was a dog, too.
Jackie Chan Adventures: One episode had Evil Brit Valmont spray Uncle with a gas that causes amnesia. Valmont says, "Fare thy well, you old goat!" Uncle, who was wondering who he was, said, "Oh! I am a goat!" So he starts walking and running around on all fours, nibbling on stuff and bleating!
Rat King from Teenage Mutant Ninja Turtles thought he was actually a rat.
In The Simpsons softball episode, one of Mr. Burns' team of ringers is hypnotised into thinking he's a chicken. He continues to think he's a chicken for the rest of the episode.

Real Life

Actually a real mental illness, called clinical lycanthropy.
According to a persistant rumor, Confederate Gen. Richard S. Ewell had occasional delusions that he was a bird.