I would link the monkey's paw to Kasen via her missing arm, but yeah, I have roughly the same feelings. They'll be connected when they can, but if not they'll just throw in something like Marisa's. They have to be in character, but that's a broad limitation rather than a narrow one.
I see it as pretty close to Mamizou turning people into animals in HM. It's not random, but it's more of a free association game than anything strict.
Edit: Kasen's title for the game says she grants wishes, which forms the rumor link, but I don't think that had anything to do with her before this game.