Alexfar,
Thank you for reaching out at Apple Support Communities today. It sounds like you’re not able to get your iPhone to make a connection to the Apple servers to save the password for your Apple ID from this prompt to update your Apple ID settings.
If your profile is correct, and you’re still running iOS 12, I would go ahead and update to the latest version of the iOS software. According to Apple’s iOS 13 notes, the iOS 13.1 update has a fix for an issue similar to the one you’re experiencing.
About iOS 13 Updates - Apple Support
If you need help with creating a backup or updating your iPhone:
How to back up your iPhone, iPad, and iPod touch - Apple Support
Update your iPhone, iPad, or iPod touch - Apple Support
Cheers.