Mentor is the name of the person to whom Odysseus (a.k.a. Ulysses) entrusted the care of his son, Telemachus, when he set out on those famous wanderings of his that we now call an "odyssey" and which took him, among other places, to the Trojan Wars. Mentor was Odysseus' wise and trusted counselor as well as tutor to Telemachus. Myth has it that the goddess Athena would assume Mentor's form for the purpose of giving counsel to Odysseus but, for many centuries now, the goddess has been unavailable for comment to confirm or disconfirm this rumor. At any rate, Mentor's name -- with a lower-case "m" -- has passed into our language as a shorthand term for wise and trusted counselor and teacher.
Until recently, in ordinary usage, the term mentor typically was used by the recipient of the counsel or teaching to refer to the person providing it. Thus, I might say that Ed Trapp was my mentor. Observers of such counseling and tutoring relationships have also been known to label certain people as mentors. Kierkegaard, for instance, has been called mentor to an entire movement. But, mentors, it seems, have not and do not characteristically refer to themselves as such.
This writer's own biases are such that he has always thought the term mentor was exemplified in a situation wherein an older, influential male took a younger, promising male "under his wing," so to speak, for the purpose of advancing the younger male's career. The older male was then a mentor to the younger one. And the younger male, of course, would be known to us as the protégé of the older man, just as Telemachus was Mentor's protégé (although it seems only fair to point out that the French language had not yet then evolved and so whatever term Mentor used to refer to Telemachus might or might not be known to us but it seems certain he did not use protégé).
In recent years, especially in the management and human resources literature, mentor, which is a noun, has become a verb as well and -- with or without "ing" as an appendage -- now refers to the patterned behaviors or process whereby one person acts as mentor to another. And, in keeping with current mores and norms, gender seems irrelevant. (This usage, however, has yet to make its way into any dictionary to which this writer has access.) It will surprise some, but not others, to learn that there are consultants whose practices center on advising the rest of us regarding the mysteries of mentoring. (They are perhaps would-be or frustrated mentors themselves.)
In sum, what has been historically an informal, unofficial, voluntary, mutually-agreeable, and self-selected interaction between two people has become a program -- an institutionalized stratagem for trying to force what probably can only come about naturally -- and a staple, if not a commodity, in the bag of tricks toted from client to client by many a consultant.
Frankly, I'm not against formal mentoring programs, but I'm not for them either. I'll pick my own mentors, thank you, and I'll pick my own protégés too.