I just re-watched TNG “In Theory” (S04E25) and Data says that the door to his quarters is set to only allow humanoids to enter or leave when Geordie finds Spot in the hall.
Whilst I get that it would be dangerous for Spot to wander the ship freely I think it’s kind of mean to keep him locked in a small room like that forever. There doesn’t seem to be any cat toys in the room but maybe the computer is capable of some sort of environmental enrichment we can’t immediately see.
It seems like the upper staff on the ship get quarters that are roughly the size of a 1 or 2 bedroom apartment. Not great, but not worse than what lots of people do IRL.
Plus maybe he sometimes takes spot to a cargohold for more active play. There would be plenty of space, lots of height, etc. And that’s all before even considering the holodeck.