Shaw denounced social snobbery and class distinction in Pygmalion primarily by presenting the aristocracy as people who can "be made" and can be spoofed by just imitating their mannerisms, and by pretending to be bigger than you are. In other words, "anyone", even a nobody, can pass as an aristocrat.
The making of a peasant girl with a cockney accent into a posh duchess is in itself the central idea of the story, where Shaw admittedly mocks the effrontery and silliness of it all. Even the fact that the girl herself in the beginning was trying to rid herself of her accent, and willing to pay for it, is a laughable action that reflects the people's mentality over the fashionable society versus those who were humble and simple.
The use of titles, the names used, the fact that there were "phonics" professors, Liza's father becoming rich after (in the beginning) pitching fights over money- all these are symbols of Shaw's bringing down the high airs of the aristocrats, showing that they may not be as unique as they see themselves.