Considering that my interpretation of a cat girl (or any other kind of “girl” for that matter) is a biological human equivalent, except the only physiological difference is the traits that identify what animal they are partially related to. This means that they are no different from humans in the context of ownership, though wether or not that is taboo depends on your native culture. (Looking at you, old traditions of old religions)
Intelligence is going to determine if they have rights. If they have mostly animalistic minds, I would imagine accusations of crimes against humanity being possible since people into bestiality is already a problematic minority, and would get worse if we go and “humanize” them like that. If we go reverse, and have humans with animalistic appearances (in a non “furry” way) it will be the biological equivalent of “cyberpunk” aesthetics, though it would raise eyebrows among the common populace. Make genetic engineering common place however, just like our tech and food, people won’t care as much about how much we take it for granted. It is a good question why we would want to create a cat girl of our own...
Make me wonder if it’s ok to clone yourself as a substitute child if you don’t have a spouse...
But back on topic.
Since philosophically, they are more animal than man than a man itself, it would be a worse problem if their instincts still kick in. Would you really want to take care of a human who cannot control themselves? We already have problems with violence (and warfare for that matter) among men, and wild animals are no different; it would be even worse to think about animal girls going into heat. Think of the children who were minding there own business until they walk along and see that by mistake. Stray cat girls (and boys, since every yin needs a yang) fucking in the streets would be traumatic.
Now, caring for a person is acceptable, so long as they are willing, so we have that down. As some people say, if they are treated as family, then it’s ok, but think about the animals who get abused and abandoned. Since humans can still be abusive towards each other, I think that this would intensify previous societal problems like racism since people default to calling animal people as another race, just like elves and dwarves, rather than a separate species entirely (so no interracial here).
Purchasing them is right out, because buying a life for personal gain, no matter how much you may compensate for it, should be considered wrong. Would you like it if you were sold by your parents under the promise that your new home will take care of you?
For the most part, I think it is wrong to have catgirl pets. I can see it being a possible gene mod trend if people can mod themselves quick and develop the appearance fast enough, though it will have a rough start.
A good question based off of all of this, is do you think the extremists of the Furries will co-opt this technology to make their wet dreams a reality, ushering in a new age of darkness and yiffing? I could see this apocalypse happening if cloning on a mass scale is possible...