There's cat cafes, and then there's also piglet cafes, otter cafes, meerkats, sugar gliders, hedgehogs, owls and other exotic animals. The legality differs from prefecture to prefecture.
I have mixed feelings towards them. Yes, you're keeping a bunch of animals alive and fed, but at what cost? Owls for instance are sometimes drugged so they won't freak out around people. Their sleep schedules are messed up because they have to be awake during the day, wings clipped, legs tethered to a post for hours on end, and they're not really eating the right things or getting the right stimulation to enjoy their owl lives. Maybe it's ok for cats and dogs because they're social animals who've been bred to live alongside humans?
Strays on the other hand are nice. They come up to you for pets and don't expect food.
Numakawa's first part of his name is 沼 and it means muddy swamp/marsh/bog, and it also means being stuck or obsessed. The manga uses this kanji for several of these chapters where someone is going through some sort of problem in their lives.