In Pokémon Diamond, Pearl, and Platinum, an NPC in Oreburgh City will tell the player that "naming Pokemon makes you feel as if they are your very own". The NPC right next to that one says "But he named our Psyduck Yellow just because of its color..."
Haunter's name in Japan is simply "Ghost" and the Ghost-type is still referred to as such over there. The names of some of the earlier Pokémon in Japan are also just whatever they're based on. For example, Kakuna is "Cocoon", Pidgeotto is "Pigeon", and Krabby is "Crab".
As a non-living example, there's also the special move "Psychic" which is a Psychic-type move.
Snake also comments on another SSBB character's "creative" naming: R.O.B. Specifically, he takes issue with the fact that in Japan it's called... Robot. He settles with calling it R.O.B. when Otacon points this out.
More Nintendo examples, from Super Mario Bros.: Yoshi, Birdo, and Toad are apparently not only the names of three characters, but of their entire respective species as well.
Unagi the giant eel as well, "unagi" being Japanese for eel.
EarthBound has the character Dungeon Man, the first hybrid of man and dungeon.
The Longest Journey brings us a talking corvid named Bird, which, as April notes, does fit. When he complains about his name, she comments that she would have given him a better, more imaginative name, which he eagerly adopts... Crow. Note that he is in fact not an actual crow, and April named him that way because he reminded her of a cartoon character named Crowboy.
Half-Life 2 brings us a partial example with Dog. Dog is actually a robot, and resembles a gorilla more than a dog, but his behavior is rather canine, and probably the source of his name.
It's also somewhat implied that he was a bit more dog-like before all the upgrades and modifications.
It is revealed in the not-quite-canonical Zelda spin-off Freshly-Picked Tingle's Rosy Rupeeland that Tingle is a Tingle named Tingle, raising a lot of questions that no one cares enough to answer.
In World of Warcraft, many hunters have this — pets start out named "Dog" or "Cat" or "Devilsaur" etc, but they can change the name one time. (Unless you get a Certificate of Ownership from an Inscriptor).
The four canine companions in Fallout 2 are "Dogmeat", "K-9", "Cyberdog" and "Pariah Dog". The third never warranted a real name from its creator, and the fourth generally doesn't keep owners alive long enough to get a persistent name. Or at least one that's considered usable in polite society.
Tom from Animal Crossing is a male cat, and Bluebear is, well, a blue bear.
Inverted in NieR: One of the sidequests involves killing a gigantic boar-shaped monster Shade named "Goose".
Lampshaded in Final Fantasy XII Revenant Wings: your resident blacksmith is a friendly Cu Sith who, when first introduced, objects to being identified as such, saying she doesn't call Tomaj "Hume". Tomaj points out there's a lot of Humes on the airship, but only one Cu Sith, which she seems to accept (or at least, that remains her identifier from that point forward).
Alisia Dragoon: Alisia's familiars are named Thunder Raven, Will O'Wisp, Boomerang Lizard, and DragonFrye. Guess what they all are.
Touhou has an example. One of the many youkai races is the mind-reading satori race. When we meet one person from said race, her name is...Satori. However, in this case, her name could mean "enlightenment" or "understanding", rather than just be an indicative of her species. Somewhat weird since we later meet her fellow satori sister, Koishi, who is not named after her species.
A later game features a nue named Nue.
Chrono Trigger gives us a frog named Frog, a robot named Robo, and a wizard named Magus (Latin for wizard). While none of those are actually the character's real name, that's what they're called most frequently. Crono himself probably counts as well, since he's a time traveler whose name means "time." Chrono Cross also gives us a kid named Kid.
I Wanna Be the Guy stars a kid named The Kid, who wants to fight a guy named The Guy. These are actual names: one of the former The Guys is the kid's grandfather, named Grandfather The Guy. The game's sequel has a lad named The Lad.
Sonic Adventure features Big the Cat, a big cat, but that isn't the worst of it. Big has a pet frog, aptly named, Froggy.