I don’t think it works that way. If I park my car in someone’s field, that doesn’t mean I own part of the field, and if I drive on a road, that does’t mean I own the road. I still own only the car.
Wait until commercial ventures and competing nations lay claim to overlapping areas of the moon or other off world real estate. Remember the European conquest of the New World and all the battles over territory? Think humans will learn from that, or repeat the past again?
Oh, we’ll probably repeat it. Time and time again, we just don’t pay attention to history.
Preferably, someone like the UN or some body manages the real estate and leases, no ownership.