Anyone can claim ownership of a moon, or anything else. A claim is just a declaration that something belongs to you. I claim the computer I am typing this on, but there is no universal force that will deter someone from making the same claim, or deter them from taking the computer by force, if they want.
Objects in Eve differ somewhat from real life objects because they can be owned explicitly. What stops you from scooping the MTU someone else launched? The owner attribute of the MTU that is explicit.
Asteroids have no owner attribute, so what CCP Fozzie said is absolutely true. Asteroids are never, that is to say under no circumstance that can occur in game, owned explicitly. It does not prohibit them from changing that in the future. Such a forward thinking stance would be phrased “Asteroids will never be explicitly owned.”
Ownership can still be implied. I paid for this computer, so the implication is that I own it. Someone paid the fuel cost and purchased the equipment to bring up a chunk of moon ore, which I feel also implies (but does not guarantee) ownership.
People can choose to respect that, or not. If an enemy also claims to own the moon ore, brings ships to ‘your’ moon mining op, bumps you off them while mining up everything, then, as they say, possession is 9/10 of ownership.