The title references the Greek myth of Pygmalion, about a sculptor who fell in love with his statue, and the subsequent George Bernard Shaw play 'Pygmalion', which premiered in 1912.
Richard and Patty compare themselves to Fred Astaire and Ginger Rogers, who in the 30s and 40s teamed up in several movie musicals that showcased their dancing talents.
Even though the malt shop is Louie's Ice Cream Parlor, as the shop's window reveals, Patty continues to call it the Shake Shop.
The mention of Rex Harrison alludes to his role as Professor Henry Higgins in the Alan Jay Lerner and Frederick Loewe Broadway musical "My Fair Lady" and in the subsequent movie musical My Fair Lady (1964). The play and the movie are based on George Bernard Shaw's play non-musical "Pygmailion," which also played on Broadway as early as 1914 and was also made into a movie: Pygmalion (1938).