I dislike permanent marks per se, so I don't mark any of my property, animate or not.

If I obtained property with another's mark, then I might decide to remarque it, add my own mark, or ignore it. If my sub or slave was so marked my decision would be based on what kind of mark, location, and whether it afforeded an opportunity to use it in play. For example I could just try to beat it off her... or use it to remind her she's mere chattel (an interesting kink)... or in humiliation play.