Grim has been called a variety of different things by the different characters: when he first appears Crowley repeatedly calls him the prefect’s familiar but Grim insists that he is not.
Rollo makes the same assumption during Glorious Masquerade.
The prefect calls him a tanuki (an animal that is similar to a cross between a raccoon and a dog, “weasel” on EN) upon their first meeting.
Several other characters call Grim a tanuki throughout the game including Crowley, Leona, Ace and Kalim, but usually to insult him.
Grim has the joke, “that’s animal cruelty…not that I’m an animal” on EN, but that is something he has never said (there is a place on EN where Crowley refers to him as an animal but this was also an addition to EN: in his original dialogue, he is again calling Grim a tanuki).
The closest we come to Grim self-identifying as anything might be a line in the prologue where he says that dragons are “what all monsters yearn to be,” but it is unclear if he is including himself among “all monsters” or just making an observation about monsters in general.
When other characters call Grim a monster (Crowley, Ace, the prefect, Deuce, Jamil), they use the English-language word “monster” in katakana, audible in the audio.
In one scene Crowley says “monster” aloud, over a hidden meaning of another word: dire beast.
And that might be what Grim actually is, with Riddle, STYX technicians, Rook and Ace all calling Grim a dire beast at different times, and Idia providing an in-depth explanation based on an analysis performed at STYX.
Twisted Wonderland Reference Note 