This is on 13.0.1. I presume others have encountered it.
Open the second window. Its custom menubar will not initially appear, as the first one persists. To obtain the menubar, click in the old menubar’s Help menu. Then you will get the new menubar.
As an alternative, you can click anywhere in the old menubar, then mouse-over “Help” to effect the change. Weird behavior either way.
Thereafter, as long as both windows remain open, the menubars will properly switch as the focus is toggled between the windows.
Per my comment in the issue, I later tested on the current (Intel) 13.1 beta, and it looks like it is fixed on Apple’s end.
If that is indeed the resolution, let’s hope it stays fixed.