There are/were 30 odd Blue Cross Blue Shield companies. Anthem is one, BC/BS Tennessee is another, Horizon is another, CareFirst is another.
Each Blue has a defined geographical area. They do not compete with each other.
The Blue Cross/Blue Shield Association, headquartered in Chicago, is the company that issues Licenses to the aforementioned companies so they can use the BC/BS Shield.
And no, the association does not own any of the other companies.
BTW just because they are located in different states doesn't mean only your state BC/BS covers your state. My dad retired from Ma Bell under a non AT&T company after the break up. AT&T bought the company back and the retirements as well. His BC/BS company was in Illinois. The last place I worked had AT&T. It was a nursing home. The company dropped AT&T and became self insured and we had a lot better coverage.