That wasn't my point. We can own physical items, land etc. Because it's a scarce resource. If me "owning" something doesn't prevent you from also "owning" it, the concept of ownership of that item is absurd.
I do understand rivalry in economics, but that's not the issue here. The issue is the need to protect individuals from abuse, part of which is the one's right to their own likeness. This likeness is not free for the taking. Now, we can call this owning the likeness, or we can call it something else, but having something, that is not free for taking, sounds an awful lot like owning something.