I'm way too much of a dominant personality to be a "pet". My mate, whoever they may end up being, might end up being my 'pet' though.... ^_^
If I were to have a "property of" tag, it'd probably say something like "Property of no one" or "Property of JC" (religious).
Generally I don't mind pet/submissive ideas if it's in play or within a healthy relationship. I've seen the unhealthy side of that, so it in general rubs my fur the wrong way.