When Rodolpho decides to show off his lovely tenor voice to his new benefactors, he makes the incredibly symbolic choice of singing "Paper Doll." It would seem that the symbolism isn't lost on Eddie. Let's check out some the lyrics and see what the big deal is. "I'll tell you boys it's tough to be alone. And it's tough to love a doll that's not your own. […] I'm gonna buy a paper doll that I can call my own, A doll that other fellow cannot steal" (1.308).
It seems to us that the words describe exactly the way Eddie feels about Catherine. The prospect of her growing up makes him feel lonely. Catherine isn't really his, and as much as he tries he can never truly have her. Also, he tries to treat her like a doll. He wants her to be a pure innocent object that he can dress up in girlish clothes and not let anybody else play with.
The song takes on new meaning later on in Act I. By this point, Eddie has done a good amount of Rodolpho bashing. When Catherine can't take it anymore, she jumps up and demands that Rodolpho dance with her. What song does she choose to put on? You guessed it – "Paper Doll." She's too weak to out and out rebel. She's not strong enough yet to say, "Whatevah, I do what I want!" Instead she puts on a song that she knows makes Eddie uncomfortable and chooses to dance with the guy who she knows is the focus of Eddie's disapproval. Catherine would get an A+ in Passive Aggressive 101.