Keep the cat, because you're not going to be able to get rid of it anyway. It has bonded with your family. That's actually how we got our newest cat. He showed up on our back deck one day as a kitten, and we played with him, fed him, and just took care of him. He'd run off, but he'd always come back the next day and such. Eventually, he started spending whole days here. Finally, after he decided to move in, we put a collar on him, but the next day it was gone. We put another collar on, and the next day he came back with a little note.
"My name is Smoke, and I live at so and so."
We went and talked to the owners and they said we could keep him because he was their son's cat and the son had moved away. Awesome, as we call him, still goes over there to visit, but he's our cat because there's no way he will leave now.
I would, however, tell the family about it. Assuming they care about their pets the way most people do, just knowing that their cat is safe and living a very happy life will make them feel better. It can be pretty painful wondering what happened to a loved one, even after so many years. I highly, highly doubt they're going to demand the cat back.