Do you think adapting homeless animals is slavery?
The animal would probably be killed if you didn't take it home. My family always "adopted" animals from rescue shelters, and they seemed happy enough - they certainly became very distressed if they perceived you to be trying to abandon them (e.g. getting into a car, or putting them in boarding kennels). And you wouldn't be directly supporting animal breeders/farmers. (I have heard some awful stories about farms that breed pedigree cats and dogs

Or poachers that steal bird and lizard eggs to sell as exotic pets?)
A cat will just leave if it doesn't like you

I don't know about dogs - it is more difficult for them to escape, and I think they instinctively want to be in a pack anyway? Caged animals and fish don't have a choice.
I personally don't want to have one.. ...Can't travel freely etc
This is true - I can't have a cat or dog because I live in a small rented shared house, and I'm not always in it!

that's how I got thinking about buying a small pet like a fish or lizard.
Is there other solution to saving animals?
It's a tough one

They shouldn't need saving!

But if mankind has domesticated certain animals, or destroyed natural habitats, I suppose we also have an obligation to care for animals? I'm not sure how this is best done - maybe it depends on the animal.
A thought: Maybe in a vegan future, people will keep pet cows and pigs, in the same way that some people continue to keep horses nowadays?
