Above all, we learn that this relationship is shaped by economic class.
To expand on that, look at how Mr. Doolittle acts in relation to Liza, compared to how Mrs. Higgins acts. Mr. Doolittle accepts his place at the bottom of the economic heap, and he (happily) resigns himself to a life free of ethical expectations. The best example of this is that he's essentially ready to sell his daughter to Higgins for five pounds. It's not slavery, but it's not far from it. By contrast, Mrs. Higgins has many more complex expectations for Henry, including hopes that he have the right manners.