While I will reluctantly overlook the use of "Mentor" as a verb (that battle is lost), I refuse to acknowledge the existence of the verb "to ment" that "mentee" necessarily implies.
As it happens, it's true that "mentor" comes not from a verb "to ment," but rather -- according to the OED -- from "the name of a character in F. de S. de la Mothe-Fénelon's Les Aventures de Télémaque (1699), after ancient Greek [Mentor], the name of a character in the Odyssey, in whose likeness Athena appears to Telemachus and acts as his guide and adviser."
But so what? "Workaholic" doesn't come from a longstanding suffix "-aholic" meaning "addicted to"; it comes from the last syllable of "alcohol." Likewise with "telethon," which I take it stems indirectly from the place name "Marathon." We can all come up with more examples (consider the various "-gate" scandals).
True, these words tend to have a mildly humorous feel, at least at first; so does, in my view, "mentee." But accepting them hardly "necessarily implies" any particular etymology. It just necessarily implies a recognition that English words come about in lots of different ways, and that stems are often borrowed from one word into another in ways that do not fit well with the source words' own origins.