Raptor Red: The eponymous character — a Utah raptor of the Red-Snout species — comes from a species with no names or real language, so the author gives her this nickname for reference. Other characters have similar "names" like "Raptor Red's sister" and "the old dactyl".
Interestingly averted in The Last Dragon. Yorsh, Sajra and Monser find a dog, and Yorsh names him Fido. Monser exclaims that that's a ridiculous name - dogs are called Paws or Patch or some such, not Fido. Yorsh later names a horse Lightning, to similar consternation. It's implied that humans are not particularly imaginative about naming, at least not with animals.
In The Animals of Farthing Wood, the Farthing Wood animals are all called by what animal they are such as Fox, Owl, and Badger; animals in the past, when things were more plentiful, had names like Lean Fox. But when we meet more animals, they're given different names of their own.
In the film adaptation, Holly explains to Paul that she just didn't feel like she had the right to give the cat a name because she didn't have any right to own the cat, instead seeing him as something more like a flatmate.
Most of the characters in The Jungle Book as well as its related stories are named after their species names in Hindi:
Baloo the bear, Tabaqui the jackal, Mang the bat, Chil the kite, Hathi the elephant, and so on. Subversions include Bagheera the panther, actually meaning tiger, and Shere Khan the tiger, actually meaning King Lion.
Then you get Father Wolf, who isn't given even that, and Grey Brother is nearly as bad.
A rare human example: Mowgli the man-cub's nickname is Man-Cub. They also call him Little Brother, which works for Bagheera, but sounds odd coming from a porcupine (Ikki calls him that in "How Fear Came").
In a particularly bizarre quasi-subversion of this trope, Mowgli is explicitly stated in the book to be named after the word for "frog", but Mowgli does not actually mean frog in any human language.
Ayla adopts a wolf pup and names it Wolf. In-Universe it's an Unbuilt Trope, though, since it's virtually unknown for people to have pets in that world.
Also, she calls her horse Whinney, which she takes to be another form of this.
Her orphaned lion cub named Baby may qualify. He was like a baby to her.
Adam's dog in Good Omens: "I'll call him Dog. Saves a lot of trouble, a name like that."
According to the narration, the Dog-to-be is of the opinion that his name will define the core of his nature (being, as he is, a magical Hell Hound servant to the Antichrist). So when his owner names him Dog rather than, say, Killer...
A The Royal Diaries book about Cleopatra mentions this, after Cleopatra's sister obtains a baby baboon: "I do love this sister, but she is not very imaginative. What did she name her little pet? 'Baboon'. That is it. Berenice might be beautiful, but she would not make an interesting queen."
In the trilogy that started the Shadowrun novel series, Sam Verner names the stray mutt he adopts "Inu", which is "dog" in Japanese.
Played with in Luka And The Fire Of Life. Presumably, the dog had originally been named Dog and the bear Bear, but they came to Luka with their collars swapped, so the book refers to them as Bear the Dog and Dog the Bear.
Justifed in James and the Giant Peach, since the characters are mutated insects who presumably didn't have names when they were normal insects, and they apparently haven't seen the need for a name more descriptive than their own species. After all, anyone can tell the difference between the only giant spider in the world and the only giant earthworm in the world, without needing to give them any kind of personal name.
In Mortal Engines there is a pet called Dog, although technically he's a wolf.
Dog by Daniel Pennac has the main character named... well, Dog. Other names had been suggested for him, but his owner Plum didn't like any of them.
In the rabbit mythology of Watership Down, there's a hedgehog character named Yona. "Yona" is Lapine for "hedgehog".
A Feastfor Crows features a septon (equivalent to a priest) with his pet dog named... you guessed it. (Dog)
There's also Shaggydog, but he's a wolf. The name is explained by the fact that Rickon Stark was only three when he named him.
Haplo in The Death Gate Cycle simply calls his dog "dog". He also refuses to call Alfred by his name for a long time, referring to him simply as Sartan (Alfred's race).
There is a tomcat named Tom in Warrior Cats: Dawn of the Clans: Thunder Rising, who eventually becomes the father of Turtle Tail's kits. Gray Wing finds it really weird that someone would name an animal their own species name.
This continues in The First Battle, where after a scuffle, Thunder is introduced to Tom. The latter introduces himself, and Thunder thinks he's dumb, saying "I know you're a tom. What's your name?"
A Song of Ice and Fire. Septon Meribald, a wandering priest in the Riverlands, is accompanied by a dog which he only calls Dog, claiming that he can't name it as the dog doesn't belong to him, and it has not bothered to tell him its name.
All the wild dogs in Darkeye are named their species name from their country of origin- i.e., "Mhumhi" means "African wild dog" in Shona.
In the books on which Longmire is based, Walt unofficially adopts a dog from a victim who died and, because he doesn't know the dog's name, calls him Dog. He lives in the sheriff's office and follows Walt everywhere.