A lot of little kids go through the "mine" phase as they learn about the concept of ownership. Cries of "it’s mine, it’s mine" fill the air at many a preschool. But somewhere between that age and adulthood, it seems that many people somehow lose that concept of ownership. Cries of "it’s not mine, it’s not mine" fill the air as people deny ownership of pets that have been living on their property for years.
The laws are reasonably clear: the "owner" of an animal is generally described as the person who owns, keeps, harbors, possesses, has custodial care of, or acts as caretaker. That means the stray or feral cat you’ve been feeding for the last 6 months is yours. And as the owner, you may be held liable for any damages that animal causes to someone else’s home or property.
Conversely, just because you find an animal, that does not make it yours. The problem with pets is that unlike a lawn mower or other personal property, an animal can wander off. Don’t assume that a pet you find has been "abandoned" or "dumped," even if you live out in the country. An owner may be frantically searching for the animal. It’s your responsibility to report any animals you find to the local authorities or animal shelter.
However, if you report the animal and make an honest attempt to find the owner, but no one comes forward, you become the owner by default if you keep the animal on your property. If you are not prepared to assume all the responsibilities that come with owning an animal, you should not be an owner. Take steps to find the animal a new home with someone who is prepared to handle the responsibility or take it to the shelter so they can do it for you. But realize that if you decide to keep the animal, it really is yours, just like the blocks you claimed for your own in kindergarten.