In my case, the "sign in to itunes store" popup was using an old email address that was my old/previous Apple Id.
After talking to very helpful Apple Support via chat, it turns out it was an old app, installed back when I was using my old Apple Id. The old app was trying to connect to the itunes store as part of some sort of in-app purchase. And the old app wasn't smart enough to notice that I had a new Apple Id in settings.
My solution was to delete and reinstall old apps (typically losing their data).
Note that the apps I deleted to fix the problem did *not* have background refresh allowed. But they were somehow running in the background trying to connect to the itunes store.
Also, it's really maddening that the "sign in to itunes store" popup doesn't say which app is requesting the access. The lack of basic info turns trivial problems like this into showstoppers.