I tend to use "mah boy", "my li'l girl", and "my snugglebunny" instead of "my pet".

I also say, "I'm Nick and Nora's person."

Actually, come to think of it -- that's the way I say it. I don't say, "I'm Nick and Nora's owner," OR "I'm Nick and Nora's guardian." I say, "I'm Nick and Nora's human."