I have this issue too, and can't buy in-app purchases because of it, or download new apps. When I'm in the App store and look at my account, it has me as being signed in with my backup/emergency email account. When I click on that to view the Apple ID, it just takes me to a blank screen.
I've tried, to no avail:
1) changing date to last year, as apparently fixed this issue in older phones. The server now checks this and boots you off before it does anything.
2) changing my payment details and getting it to re-auth them. This seems to make the ID fine in the Settings area, but when I go back into the App Store, it fails again and I have to re-authenticate, at which point it gives me the "Could not sign in" message. When I go back into the App store afterwards, it has the emergency/backup email address there again.
This is very frustrating!