Shaw’s play is based on Greek mythology and the story of Pygmalion (written by Ovid). Pygmalion is a king of Cyprus and a gifted sculptor. He is perturbed by the lustful and immoral characters of women of his age and place and feels displeasure in their midst.
In his disaffection, Pygmalion was inspired to sculpt his ideal woman. After finishing, he fell in love with his own creation and begged the powers above to breathe life into it.
He named her Galatea and fantasized about having a real-life romance with her. His wishes are answered in the end and the sculpture comes alive. As the legend goes, they end up marrying and having a baby daughter.
In the play, Higgins is the sculptor and Eliza is his creation. However, contrary to expected emotions of romance and companionship, Higgins and Eliza do not develop such bonds.
Higgins is not interested in marrying as his ideal woman is his mother Mrs. Higgins. To him, Eliza is just a challenge and an opportunity to help a girl in need.
From the perspective Eliza, she never receives the devotion that Galatea receives from Pygmalion in the legend. Thus, she too never develops a sense of affection and intimacy that is seen in the story of Pygmalion and Galatea. She ends up choosing Freddy who does worship her and wants to build a home together.