In C'mon Digimon, Makoto Abe and Kentaro Kamon are the only ones who actually give their monsters names. Everyone else merely calls them what they are (Meramon, Deathmon, ect). Downplayed in Digimon V-Tamer 01, where most plot relevant monsters do have names and out right subverted in Arkadimon's case, as that is it's name. On the other hand, Bun's species name is only implied but never outright confirmed while Arkadimon, Pal and Pul's species names are never given(well, Pal and Pul were called Petitmamon years after V-tamer was done). The Digimon as whole played this totally straight after C-Mon and V-Tamer were done though, despite some monsters being fluent in human language (unless you count Chaos Lord and Neo Crimson from Digimon World 2. Once again, species names never revealed)
Oddly, in The Big O, there was a cat named Perro.note Though, in Japanese, "pero" is also the onomatopoeia used for cats licking themselves (Japanese has no double-R), giving an alternate explanation for the name.
In Gurren Lagann, we have Simon's pet pig-mole, Boota (or Buta). Buta literally means "pig" or "swine" in Japanesenote e.g. Kurenai No Buta, which means "Pig of Crimson", or "Red Pig".
A partial example of this is found in One Piece, where there are at least 2 Fishmen that are named after the fish they're based on. Their names are Jinbe and Daruma. The former is a whale shark Fishman (In Japan, whale sharks are called "jinbee-zame") and the latter is a cookie cutter shark Fishman (cookie cutter sharks are called "daruma-zame" in Japan).
This is almost omnipresent in Pokémon. While the species names are either Pokémon Speak or Verbal Tic Name, it's a little weird that nearly every pet Pokémon is named after its species. This is most likely due to marketing reasons (viewers would get confused and think that the given nickname is what the whole species is called).
In Pokemon Special, trainers who bother nicknaming their Pokémon usually give nicknames that barely deviate from the species' name anyways. In fact, White is the only nicknaming Dex Holder who gives her Pokémon regular names, albeit slightly punny ones. In an interesting variation, X gives his Pokemon nicknames based off of their French names.
The heroine of Princess Tutu is a duck who can turn into a human girl. Her name is Duck. Often she's still referred to by her Japanese name, Ahiru... which is Japanese for "Duck."
Ranma One Half: Subverted by Kodachi's pet, Mister Turtle, a crocodile.