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Passing the Torch, just a protagonist variant.
Not really. Passing the Torch is in effect when the mentor wants to retire and let someone else do his job. This is NOT what I'm talking about. What I'm talking about is when the mentor in question does not want to step down from his position, but still gets an apprentice/pupil. For exemple, take Goku and Gohan from DBZ. Goku took his son under his wing during the Cell saga to teach him stuff like how to become a super saiyan, but he didn't plan to retire at all. In fact, he continued long after that and is still continuing today.
WAPOW
You might want to look at The Apprentice, Master-Apprentice Chain, Student–Master Team, and related tropes.
I really don't see why the mentor archetype should inherently exclude the protagonist, so much so, that it would require its own trope instead of simply ascribing them as The Mentor or its more relevant subtropes.
Because this specific instance/variant happens many times more than once. The mentor archetype trope has spawned many offshoots. And yes, before anyone asks, I've looked at not only the entire Mentor Index to make sure that this specific variation wasn't present in it, but I've also looked at the tropes that you guys told me to look at. Sure, they come close, but they don't really match the variation in question...but still, I appreciate the help. Thanks, guys. :)
WAPOW
Is there a name for the trope where the main character is or becomes a mentor for another character or takes a student/pupil/apprentice under their wing in the story?
Example 1: Alice is the main character and she decides to take Bob as a student to pass on her knowledge. Example 2: When the story starts, Alice is the main protagonist and she has a student named Bob that follows her on her adventures.
Note: For the trope to work, the character in question HAS to: -Be the main protagonist. -Have a student/pupil/ward when the story starts or has to take a student/pupil/ward under their wing later on in the story(still counts if it happens at the end).
In addition, the trope does NOT work if the character in question is a former main character or isn't the main character of the story(unless it's a spinoff starring another character) It has to be a main character in some way or form no matter what.