It is quite acceptable to deem Eliza's first test in Act 3 as a "precious bit of drama" in many respects.
In this Act we find Mrs. Higgins at her in-house, around 4 to 5 in the afternoon expecting visitors who have send in their calling cards to visit her that afternoon. In this case, it was the Eynsford-Hill duo of Mrs. Eynsford and her daughter, Clara. Mrs. Higgins, like Mrs. Eynsford, belong to a sophisticated and conservative middle class which, in Victorian England it would translate to our American equivalent of an upper-middle class. This is the reason why Higgins chooses to test Eliza's diction and poise in that environment.
The only person who is out of place there, however, is none other than Higgins. He is pedantic, abrupt, cynic, mean, sarcastic, obstinate, selfish and indifferent to the presence of others. He will speak, say, and act whichever way he pleases, making his mother quite uncomfortable.
In comes Eliza Doolittle. The moment she enters the scene the description is so comical that one can almost see her, a flower girl, posing in the most imperious, exaggerated, and insufferably snobbish way. It almost brings the audience back to every comedy of manners in which one character stands out for its extreme posh behavior. A character that stands out is Lady Bracknell from Oscar Wilde's The Importance of Being Earnest.Being that Shaw and Wilde are contemporaries on stage, one cannot avoid sensing each other's influences in the creation of their character.
This aside, Eliza's description reads:
Speaking with pedantic correctness of pronunciation and great beauty of tone] How do you do, Mrs. Higgins? [She gasps slightly in making sure of the H in Higgins, but is quite successful]. Mr. Higgins told me I might come.
So here we see Eliza, inhaling and exhaling her H sounds, cutting her sentences to a bare minimum to demonstrate familiarity with the current situation, and imperiousness to separate herself from the rest..as if she already were a Duchess.
However, although at the beginning things seem to go OK Eliza begins to slip in semantics, not so much in her intonation. Her strength in expressing frustrations is so strong that she cannot keep it in check even under monitoring. This is precisely when Freddy finds her to be a unique and precious individual aside from everything else. Here is Liza explaining how her father is an alcoholic but that the drink has not really harmed his life all that much.
LIZA:Not a bit. It never did him no harm what I could see. But then he did not keep it up regular. [Cheerfully] On the burst, as you might say, from time to time. And always more agreeable when he had a drop in. When he was out of work, my mother used to give him fourpence and tell him to go out and not come back until he'd drunk himself cheerful and loving-like. There's lots of women has to make their husbands drunk to make them fit to live with. [Now quite at her ease] You see, it's like this. If a man has a bit of a conscience, it always takes him when he's sober; and then it makes him low-spirited. A drop of booze just takes that off and makes him happy. [To Freddy, who is in convulsions of suppressed laughter]Here! what are you sniggering at?
The new small talk. You do it so awfully well.After Eliza leaves, her imprint is official in Freddy. Yet, she is able to be herself and talk the way she truly wishes to express herself. For this, she charms most, if not everyone, she comes across with.