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)
Subverted in A Certain Magical Index when one of the Misaka sisters finds a stray cat. When Touma asks what she wants to name it, she responds "dog".
A partial example in Fullmetal Alchemist. The villains of the series are the Homunculi, and the original one, who eventually becomes the "father" of the others, was called Homunculus. It's a partial example because he was also known as "the Dwarf in the Flask" (and is addressed using both names together, as "Dwarf in the Flask, Homunculus"), and once he's created his offspring, he's referred to as "Father" by most characters. It's worth noting, however, that Hohenheim always calls him "Homunculus" and/or "Dwarf in the Flask"note Because Hohenheim is the only character old enough to have known Father before he became Father., and once his "children" have been defeated/killed, other characters do so as well.
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.
One of Rias' Pieces is a junior Catgirl named Koneko, Koneko being Japanese for kitten (or literally, small cat). Justified, as it's an alias given to her by Occidental Otaku Rias. Her real name is Shirone.
There's also a Welsh dragon named Ddraig. Of course, given the setting it's plausible he isn't named after Welsh dragons - Welsh dragons are probably named after him.
This is how the animal characters are named in Jack and the Witch, an anime movie released by Toei i the 1960snote Only the dub released by American International Pictures gives them actual names (such as the mouse being called "Squeaker".
A partial example of this is found in One Piece, where there are at least two 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).
Hakase in Nichijou sort of counts, as "hakase" is the Japanese word for "professor", and she was never revealed to have a real name.
This is almost omnipresent in the anime. While the species names are either Pokémon Speak or Verbal Tic Name, it seems 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). The number of nicknamed owned by main characters can be counted on one hand: Misty's Luvdisc named Casrein, James's Growlithe named Growlie and Lillie's Vulpix named Shiron/Snowy. Apart from that, nicknamed Pokémon are mostly owned by very minor or one-shot characters.
In Pokémon Adventures, 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 ½: Subverted by Kodachi's pet, Mister Turtle, a crocodile.
In SuperLovers, "Tanuki" who's called "Raccoon" in the translations. It's a little complicated, as "Tanuki" is a Japanese word meaning either "badger" or, usually, "raccoon dog", the latter of which looks similar to a raccoon (but is actually a canine), so, this is more of a "close approximation in translation" case (i.e, the tanuki in the Eng. dub of Pom Poko were called "raccoons").
Slime Life: Slime is a slime who introduces himself to Darul as such. There is no indication given that he has any other name.
In Toradora!, Ryuuji's parakeet is named "Inko-chan". "Inko" happens to be Japanese for "parakeet".
In Wild Life, Tesshou's dog is named Inu, which is dog in Japanese. He also befriends a bear, which he calls Kuma, that is bear in Japanese.
In Yo-Kai Watch, all yo-kai go by their species names. In the games they can be nicknamed but the anime opts out of this. We know thatJibanyan's name was "Rudy" in life, but he strictly goes by "Jibanyan" as a yo-kai.
Yu-Gi-Oh!: The majority of monsters are just named after what they are, and only less than half of them have proper names, such as Osiris/Slifer, Raviel, Wisel, Kibou'ou Hope/Utopia or Kali Yuga. For example Black/Dark Magician is always called Black/Dark Magician (even though Mahad might be a valid name for him) and Blue-Eyes White Dragon is always called Blue-Eyes White Dragon (or simply Blue-Eyes for short). This trope is particularly common with Dragon-Type monsters, with only a handful of those having proper names, e.g. Blaster, Ultimaya Tzolkin or Ignister. Most characters in the various anime and manga series shorten the names of the "nameless" dragons by simply leaving out "Dragon" or other words, for example "Blue-Eyes" for Blue-Eyes White Dragon, "Ultimate Dragon" for Blue-Eyes Ultimate Dragon, "Red-Eyes" for Red-Eyes Black Dragon, "Stardust" for Stardust Dragon, "Shooting Star" for Shooting Star Dragon, "Ancient Fairy" for Ancient Fairy Dragon, "Red Daemon's" for Red Daemon's Dragon, "Odd-Eyes" for Odd-Eyes Pendulum Dragon, "Odd-Eyes Rebellion" or "Haohkokuryuu" for Haohkokuryuu - Odd-Eyes Rebellion Dragon, or "Starve Venom" for Starve Venom Fusion Dragon. Those shorten names are always used as if they are actual names.