What effect does Shaw get from referring to them all as The Mother, The Bystander, The Flower Girl, The Gentleman, and so on, instead of by their names.

I think Shaw wants to dehumanize people and their personal identities. England in the Victorian era was obsessed with class. People were relegated to their class and what they did for a living. The rich would place social stereotypes on various people to give their existence meaning. The person behind the job or class group, like Eliza, was ignored.