How to describe an animal lover who is against pet ownership

Solution 1:

animal protectionists

Animal protectionists will give homes to animals, but not regard them as pets. For example, I have had cats for nearly 30 years, and support several organizations that find homes for homeless animals. I am not the owner of my cats, but the guardian of my cats. (Sometimes I think of myself as the unpaid employee of my cats.)

I found this term by starting with the suggestion of @Dan Bron: animal rights advocate. From Wikipedia.

Advocates of animal rights as well as activists for animal liberation hold the view that to deny the most basic needs of sentient creatures—such as the avoidance of pain—to non-human animals, on the basis of species membership alone, is a form of discrimination akin to racism or sexism. Many animal rights advocates argue that non-human animals should be regarded as persons and members of the moral community whose interests deserve legal protection.

Within this group, there is a spectrum ranging from those who avoid being the cause of death of any living thing (vegans aren't even at the extreme end) to those who are what the Wikipedia article calls animal protectionists:

A distinction persists within the movement—based on the utilitarian/deontological divide—between those who seek incremental reform, a position known as animal protectionism, and those on the abolitionist side, who argue that reform that aims to regulate, rather than abolish, the property status of animals is counter-productive.

In summary, I doubt that you will find a single word, or even a simpler phrase than the one I have suggested.

This is not the place to define the spectrum of animal rights advocates further, although PETA and ALF must be mentioned. I suggest you formulate a question that would be on topic on the Philosophy site https://philosophy.stackexchange.com/ or the Pet site.