This is where the notion of ownership becomes retarded. We're going to commit the same acts of stupidity on the Moon that we do/did on the Earth if we don't change our ways. Territorial disputes account for what...over half of all wars in history? Yeah, those are great parts of our history, where we killed each other over dirt (or the resources on or below said piece of dirt). Of course, that's no better than killing each other over fairy tales, mythos, or faith I suppose.
Sure, let's do the same stupid things on another solar system body by establishing property, ownership, false borders and selfish nonsense...idiotic.
No one owns the Moon, just like no one owns the Earth. And the notion that it's human nature to assume control and fight for 'things' is asinine. We are TAUGHT those values, they are not born into us. You can teach people to share and get along just as easily as you can teach them to be selfish and fight for pieces of dirt. Unfortunately humanity isn't civilized enough to teach the former of those two...yet.
I'd like to assume we would learn from our mistakes and not take our negative baggage with us when we start to expand humanity into the stars. I guess that's too much to ask once you bring lawyers into the mix.