You're missing the point. The USA may be able to say that they own the moon, and even get everyone on earth to agree to it, but without a permanent base on the moon, and a way to defend it, they can't enforce the claim.

Let's imagine that the whole of a small country, like France, is magically teleported to the moon. (With a protective air bubble, and the necessary resources to survive and defend it.)

The entire population of earth might still agree that the moon belongs to the USA, (as they could be coerced by military action and trade agreements), but the entire population of the moon would say the moon belonged to France - or the Moonians as they might call themselves - until the USA becomes motivated to put a competing presence there.

Flags are an effective means of designating territory on earth because they represent the nation whose power is backing up the claim. The flag on the moon is not backed up by the US Navy or Airforce, but instead with the knowledge that they last put three people there, for under a week, 41 years ago.

This land is my land.

...

We also own the moon.