Class structure is at the heart of this story which relates to your question. The social hierarchy is an unavoidable reality in Britain, and it is interesting to watch it play out in the work of a socialist playwright. Shaw includes members of all social classes from the lowest (Liza) to the servant class (Mrs. Pearce) to the middle class (Doolittle after his inheritance) to the genteel poor (the Eynsford Hills) to the upper class (Pickering and the Higginses). The general sense is that class structures are rigid and should not be tampered with, so the example of Liza's class mobility is most shocking. The issue of language is tied up in class quite closely; the fact that Higgins is able to identify where people were born by their accents is telling. British class and identity are very much tied up in their land and their birthplace, so it becomes hard to be socially mobile if your accent marks you as coming from a certain location.