As a counterpoint, for the few games I do play, I far prefer to use my real Facebook identity. Words with Friends in particular is most fun when I'm playing against my real friends. I also really like QuizUp.
Dinky little mobile games like words with friends fall into a different category than pc games. They are time-killers, and your gaming identity is not tied to an avatar that kicks ass in a virtual world. Your real identity for mobile games is simply to make finding someone to play the dinky game with easier on your phone.