In Pokémon Diamond, Pearl, and Platinum, an NPC in Oreburgh City will mention 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 in Gratuitous English. For example, Kakuna is "Cocoon", Pidgeotto is "Pigeon", and Krabby is "Crab".
As a non-living example, there's also the Psychic-type move called "Psychic".
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.
This returns in Smash 4, only this time, it's Pit and Palutena pointing it out. Pit attempts to come up with a better one.
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.
In the Yoshi series there's a dog named Poochy that the player can ride over spikes.
Unagi the giant eel as well, "unagi" being Japanese for eel.
In Paper Mario: The Thousand-Year Door, there are 7 MacGuffins you must collect called the "Crystal Stars". Each of them have the names of gemstones in their names, such as the Sapphire Star and Garnet Star (and then, there's the Gold Star), but for some reason, the last one is simply called the "Crystal Star".
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 the Fallout games, "Dogmeat", then "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.
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. However, her spellcards, dialog, and the fact she's the only known nue in the series suggest that she might be the 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 note their real names are Glenn, Prometheus and Janus, 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 and an anthropomorphic turnip, mushroom and wildcat named Turnip, Funguy and Lynx, respectively.
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.
In A Fine Day for Reaping Death's skeletal horse is named Horse. (According to a parenthetical comment, he isn't exactly known for his imagination.)
The homebrew DOS game Rallo Gump was created by a small game development company appropriately named "Homebrew Software."
In the Arfenhouse series, one of the major characters is a dog literally named DOG. This is lampshaded in the prologue of the third game, where a normal-talking Commander, trying to think of a suitable formal address, has to settle for calling the character "Mr. Dog." There's also the antagonist of the first two games, a cat named Evil Kitty (who later becomes Good Kitty), and a Pikachu named PIKACHU (or possibly PIAKCHU, no thanks to the deliberate prevalence of typos in the text).
Rabio Lepus stars a mecha rabbit called Usagi, a name which has to be written in romaji to distinguish it from the common Japanese word for rabbit.
Inverted in Wildcard by Ace and Kris' dog, who's named Katt.
Fairly common among the young children who name their pets in Webkinz. In-game, there's "Ms. Birdy", a bird who runs the Adoption Center.
In Rainbow Mosaic 3: Love Legend Yoichi marries a kitsune called Kitsune.
Moshi Monsters has six adoptable species: Poppets, Katsumas, Furis, Luvlis, Diavlos and Zommers. The movie has six protagonists, one of each species, all named after their species. The game also has a minor character who's a minibeast named Bug.