Shigeru Miyamoto, the character's creator, used the English word "donkey" to evoke the animal's stereotypical stubbornness. (There was no translation error; the game has the same name, "Donkii Kongu", in Japan.) "Kong" is because, in Japan, "King Kong" has become a generic term for any large ape.
Thus, Donkey Kong is a stubborn large ape.