A variety, of forms, of course but often a kind of march form. Think of "After You've Gone" or "Darktown Strutters Ball." Two loosely-related halves. (I'm ignoring the existence of verses in all this because, um, I want to.)

The earliest recording of an AABA chorus I've found is "Whistle It" sung by Ada Jones in 1912.
https://www.youtube.com/watch?v=cEy6AuLufW0

I don't think this song set the world on fire. The first well-known (to me) song I found is "Pretty Baby" sung by the ubiquitous Billy Murray in 1916.
https://www.youtube.com/watch?v=TMpOE_wMADI

But my knowledge of sheet music -- which was very much alive -- is basically nil. And, of course, I'm searching songs that, for some reason, I've got in iTunes. This may not be a neutral sample.

Still…