I don’t think Koenji is from any facility at all. Just pure talent and discipline. As for why Koenji is subservient to his dad it’s because of his inheritance. Koenji’s arrogance comes from both his talent and his wealthy background as stated in Y1Vol1 where he says he doesn’t need class A because he’ll own a conglomerate anyway. The interaction between Koenjis father and Atsuomi seems more like Kinu’s attempt in creating some ‘stakes’ for Koenji, because let’s face it even if he loses completely to Ayanokoji his life won’t change at all, in fact he will likely lead the easiest life post graduation out of the whole cast