They are The Government; they can do what they please :-)
Does the U.S. rule space?
Not yet, unfortunately, but the FAA does 'rule' (oversee) public and private air travel engaged in by U.S. companies and launching from US soil.
Is the FAA the only authority on space travel on earth?
No, but if you'll read the article, you'll notice that these rules apply to U.S. -based corporations and flights from US soil.
Other international agencies will likely issue rules for other corporations and for flights from other nations.
That was obvious. It's the headline. It implies that the FAA is the final authority.