Crazy mother - hopefully this gets addressed later on, maybe in a flashback. Having a mum like that probably lead to a terrible life growing up, which would shape his personality in some way, which could still be present in his new life.
The Yakuza girl - no panic, and no visible concern to whoever was with her before she ended up in a river dying. Unless whoever was with her was the reason she became like that. Maybe a traitor, but it wasn't mentioned.
The Yakuza - called a Yakuza, has japanese looking thingimajigs around the place, but their outfits look western. Didn't seem to be in full blown panic after the girl disappeared. Maybe they didnt know and trusted she was safe.
Conclusion - idk. The girl seems to be cold in personality, but quick to get attached to whoever she chooses. Either naive, or can see through peoples hearts.
The mc is just the usual isekai protagonist
