The moon is not as far away as some things, but it's definitely farther away than your nearest McDonalds or the bottom of the ocean or, you know, anything on Earth at all. Basically it's not easily accessible. So it would be pretty presumptuous for a person or group to say that they owned a certain part of it. Or all of it. But it's kind of complicated to figure out what would happen if anyone tried.
Vsauce explains the surprisingly involved history of space ownership, and walks us through international treaties, the logistics of trying to defend the moon from the rest of humanity as an individual, the real estate nightmare that is satellite orbits, the difficulty of defining "fleshly contact" with extraterrestrial bodies like the moon, and so on. The 1967 Outer Space Treaty says that everything in space is the "common heritage of mankind," which after watching this sounds optimistically equitable.