A rare human example: Kid the kid in Dick Tracy. By the end of the film he chooses the name "Dick Tracy, Jr." but is still called "Kid" out of habit.
The title character himself counts, since he really is a "dick" (as in the outdated slang term for a police detective). Only one character in the movie actually calls him "Dick" - and since this character is the Big Bad, it's arguably a Stealth Insult. The other characters call him either "Dick Tracy" or just "Tracy." In fact, it's possible that "Tracy" is his first name.
Shotgun Stories is centered around three impoverished brothers living in rural Arkansas called Son, Boy and Kid. Their father was an abusive drunk. After he got sober and found religion, he fathered several more sons with normal names.
The Aliens from the Alien franchise. The closest they have to an in-universe name other than "Aliens" is "Xenomorph", which means "Alien form". In Alien: Resurrection, Call refers to the species as "Aliens", both in talking to Ripley-8 and when calling them to attack the villain over the intercom.
In the comic book spinoffs, they're sometimes referred to as "Linguafoeda Acheronsis", but that probably doesn't count.
In Cowboys & Aliens, nobody bothers to give the dog that follows Lonergan around a name. Except Emmett, who calls it "Dog."
Basement Jack: "You named the dog Dog?"
Turned on its head with Tarzan's pal Cheetah, the chimp.
But played straight with the introduction of Boy, Tarzan and Jane's adopted son.
This is actually an improvement over "Elephant," which is what Tarzan originally wanted to name him.
The sequel to Thanks Killing revealed the killer turkey's name is Turkie.
Holly in Breakfast at Tiffany's has a cat called Cat. She explains that "The way I look at it I don't have the right to give him [a name]. We don't belong to each other; we just took up by the river one day."
In Oculus Kaylie has a dog called Dog that she brings with her as food bait for a malevolent entity residing in a haunted mirror.
Reds: Reed and Bryant have a dog named "Dog", and Louise says they have to come up with a real name. A later scene reveals that they eventually named the dog Jessie.
Joey has a baby kangaroo called Joey.
In The Deserter, Capt. Kaleb is accompanied by a large wolf dog who has its own grudge against the Apache. Kaleb says his name is 'Dog'. He also claims the dog isn't his and just follows him around for the chance to kill Apaches.