I am being pedantic, but I hate the word mentee. I am not sure if it was made up, but it sounds like it. The proper word IMHO is protege.
Mentor is of greek and latin extraction directly, there's no -er suffix added to a verb, it is a noun outright.
It is ... inappropriate to match mentor with mentee because the former doesn't have the -er suffix.
Obviously all words are "made up" but mentor/mentee doesn't follow existing patterns, it was just created by people who don't know or care and has persisted.
In the Odyssey, Mentor was a sort of advisor to Odysseus and that is presumably where the word comes from.