good question. i have no idea
my approximate understanding of izuku’s background (so far) is that he disappeared when he was like? eight years old? when katsuki finds him again, izuku has no memory of katsuki or his mom; he seems to find it absolutely hilarious that katsuki thinks he’s someone called ‘deku’ (“It means useless? That’s hilarious. I’ll take it.” and he gleefully introduces himself as deku to everyone associated with Katsuki thereafter).
not sure if he actually is part of the villain alliance and its nefarious schemes. mostly he just seems to be here to have a good time. probably would help out the person who asked him for help first. likes people on both sides. izuku probably functions more like a wild card than someone firmly in the villain alliance camp. fond of shigaraki, though, even though shigaraki seems to loathe him with every fiber of his being.
i think yuuei and class 1-a all know he’s categorized as a villain, but since no one can figure out what to do with him and no one has managed to hold him in a cell for more than 3 hours yet, izuku ends up freeloading at yuuei a lot. aizawa stops chasing him away at every opportunity after he sees izuku giving students tips on fighting in crowded conditions.
( “so, what kind of villainous things have you done?”
“mm, i don’t know. mostly i just do what i want to do, like everyone else.”
“have you ever killed anyone?”
“no.”
“stolen stuff?”
“haha, yeah, that’s my favorite.”
“destroyed public property?”
“yeah, a lot.”
“terrorized the general populace?”
“not on purpose. does vandalism count?” )
as far as villain!izuku goes, i think either a) his father, midoriya hizashi, kidnapped him and raised him a villain; b) his father is all for one, but izuku is a free spirit and sort of diverged from AFO’s goals, so AFO just turned his attention to shigaraki instead; or c) AFO picked izuku despite being unrelated to him, and izuku also skipped out on his villain alliance plans anyways.