T.S. Eliot once wrote that "bad poets imitate; good poets steal." I imagine the same can be said of composers; after all, there are only so many notes.
Sir Andrew does it. There is a bar of Puccini, a couple of bars of "School Days", and so forth. But then, as Eliot wrote about,he takes that element and makes it something of his own. I could probably write a completely original song in which not one bar was original.
If Styne did borrow a bar or two from Berlin, he put his own twist on it and made a new song. Nothing wrong with that.
That makes asmuch sense as saying, Sondheim copied that "I had a dream, a wonderful dream" motif for "it started out like a dream" in "Good Thing going".
And the more I think about my presumptions, that might even be a true subtext. Oh my. This even would work.
I'm right about it musically though. And it still would work as for the meaning. "Good Thing going" is about a dream that's ended by then. Sort of Mama Rose has given up.
Well, it wouldn't be too surprising if this motif was just still somewhere on Sondheim's mind. After all he sure had heard it a few thousand times in the late 50ies.
Actually, Sondheim did a little semi-stealing from Styne. In "Anyone Can Whistle" there's a song called "See What It Gets You" that sounds a lot like the "All your life and what does it get you..." section of Rose's Turn.
In this case, of course, it is the motif that is the issue, whether used in the overture or in the song, Gypsy4. It's just that in the overture it's highly prominently featured.
Anyhow, it is probably just a clever way of paying hommage to other composers and not pinching.