Some people like to interact with street animals like cats/dogs. By feeding them playing with them but doesn't adopt them because of liabilities. So the question regarding them going to be allowed or not?
Although "pet" means a domesticated animal who is kept for as a companion for amusement or companionship, I think the questions about the animals you gave example of, should be allowed. Just because a person for his personal reasons don't/can't want to take responsibility, does not mean he does not love the animal. And if you love an animal, you can do all things for him that you would have done if it was your domesticated "pet". So, I think they should be allowed.