Companions are always good as far as Ritsuka's concerned, which is probably part of why she attracts so many. Though she doesn't keep pets, herself (Fou belongs to himself) for what's likely similar reasons to Jinwoo.

"I'm guessing it can get unpredictable or dangerous?"

She knows extremely little about the Black Order, but she can guess. And if that cat wants to be held, it's welcome to it. Ritsuka's used to handling a small animal climbing on her, even if Fou usually goes straight for the shoulder perch.