The 1967 Outer Space Treaty mentioned in the article: https://en.wikipedia.org/wiki/Outer_Space_Treaty
Noone is claiming celestial bodies... it's just that if you mine something in space, and bring it to Earth, you can claim it (i.e. when it's already on Earth). The bits left unmined in the space are still unclaimed.
On Earth, you can casually assume that effectively every cubic centimeter has some life in it. That assumption, and other similar ones, must be examined when discussing space.