I completely forgot you can sign out of the App Store separately.
After signing out from the App Store and trying to download a new app, I also get the popup in your screenshot with no option to create a new Apple ID. Now, I don’t know if this is because:
A) I’m still signed in to my Apple ID account under iCloud
B) There’s some temporary problem with creating new accounts through this popup
or
C) In iOS 11 new accounts can no longer be created through this popup
What I do know is that this is not because of the Apple ID creation limit I mentioned originally - no Apple IDs have been created with this particular device.