The word mentor derives from Greek mythology. When Odysseus left to fight in the Trojan War, he left his friend Mentor in charge of Odysseus’s son Telemachus. When the goddess Athena visited Telemachus, she took the form of Mentor and advised him to resist the advice of others and to go in search of his father.
In modern usage, a mentor is an advisor and teacher to another person, usually someone younger and less experienced. In other words, we are calling that person the name of the historical Mentor, just as we call someone who gives presents a “Secret Santa,” or a smaller person (or company, etc.) going against a much larger entity a “David” going against a “Goliath.”
The backformation word mentee treats the word mentor, with its –or ending, as if it means “one who ments.” Thus a mentee would be “one who is mented.” This is clearly not the case. Although a contractor is one who contracts (for a particular job,) an author is not one who “auths.” Employers employ employees, detainers detain detainees, but mentors do not ment mentees.
There exists a perfectly good word for the role of someone who a mentor mentors – protégé. The dictionary defines protégé as “a person under the patronage, protection or care of someone interested in his career or welfare.” In other words, the exact description of one who has a mentor.
It takes the twisted mind of a sociopath or a bureaucrat to create a horrendous, misbegotten atrocity and encourage its use to replace a lovely, useful, and faithful word such as protégé. Those who repeat the process are just as guilty of verbal crimes against humanity as the creator (with its –or ending.) For the sake of your soul, use the correct word.