I think it's something to do with the animals owning us, because we're working to clean up after them. I'm not sure it's the strongest argument against using the phrase "own" an animal. Sure people own animals. People also used to "own" slaves. Now depending on your definitions of property, and that can become a very lengthy--and separate--side debate, ownership of a living entity is currently legal (so long as that entity isn't human or, for a California example, a ferret).
I know the PC description is pet guardian, but it's all simply in how you look at it. Sure, I'm my cat's guardian. If the vet asks me whether or not the cat is mine, I will say yes, as I have a claim on my cat, whether true ownership is assumed or not is not my problem. I will have a "claim" on my children some day, but I will not "own" them. I think many people confuse this issue even with children, thinking they "own" their kids until they're 18. Don't take that comment too far. I'm simply saying that some people have a mentality that doesn't respect other living creatures as independent, sentient entities. I dislike that type of thinking for both people and animals.