Like some other people, including myself, Freddie has a knack for languages. This can be seen with how fluent he speaks either Spanish, French or Arabic. I myself for example, as a German, am able to speak either American or British English at will. To make sure though, it is possible that production provides a linguistics coach for Freddie, if needed. Also, it is currently quite common that british actors speak American English in shows and movies, i.e. Hugh Laurie in House MD or Gary Oldman in Leon. Finally let's not forget John Hillerman, who as a Texan delivered quite a fair british English as Higgins in Magnum P.I.

Как всё сложно в этом мире. Английский язык такой разный, а не универсальный. И почему сериал называют шоу?

It is kind of odd, but it's probably just because he's the lead actor, one of the script writers, one of the executive producers, etc. It's not officially HIS show, but there's no denying how important he is to the show.