Sadly, every veterinarian has experienced the disappointing and unpleasant sight of coming to work and finding an animal left behind by it's "owner."
With that in mind, I came across an article by Raymond Lam who takes a sharp Buddhist approach to animal "ownership." Some of the issues he raises are provocative but correct and necessary:
1) Humans need to view the world through the animal's eyes not just theirs. Buddhist focus on ahimsa or not doing harm to any being "should be broadened to include our non human others when “owning” a companion being. Radical empathy, to see the world through the animal’s eyes, is needed: how would I want a family to treat me should its members suddenly have to move to a new town? How would I want a family to treat me should I fall ill with a difficult illness? Most importantly, could I ever imagine a scenario in which I was benefited by being abandoned? The answer, of course, is emphatically not."
2) Humans should never adopt an animal just because they're bored. Speaking personally, he writes: "I resolved to not keep another companion being until I’m in a position someday to give it most of my time, effort, and attention. Even the idea of bonding it with a second companion being, to stave off boredom and unhappiness, is not enough, at least to me."
3) Humans need to consider deeply and fully the responsibility of pet ownership. "The main reasons given for abandonment are: moving elsewhere in the city or to another country; the apartment or housing complex does not allow companion beings; a lack of time; or behavioral problems. In essence, the reasons consist of priorities that did not take the companion being’s full and healthy life span into account—a healthy dog, for example, can be expected to live about 10 to 12 years."
Read more here: https://www.buddhistdoor.net/features/the-dignity-of-companion-sentient-beings