Architecturally I think this style is called "How to spend as much money as possible on something that is still just a big shed in the desert".
Seriously though, was there any real need to build a "spaceport"? Wouldn't White Knight/Spaceship 2.0 equally capable of operating out of any decent-sized airport, if you make sure you arrange priority landing for the spaceship? (Does Spaceship Two land on a runway?)