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I was never really sure. I can't self-insert because I don't look exactly like the boy or girl trainer but considering you name them and play as them then by all means they're basically (You). But then you have media like manga, spin-off games, and even some shows trying to give them canon personalities for whatever reason and that throws a wrench into how much of a self-insert they really are so then I'm tempted to say no. But then again, Pokemon is, first and foremost a shitty video game franchise so if the game says they're a self-insert then that should be considered above all else, shouldn't it?
I suppose in the end it's whatever the individual wants.