Old Apple ID that is not an email

Long shot - have you tried using the email address that is associated with the Apple ID?

Does this work?

Email, password, and security questions

1. Go to your Apple ID account page and click "Forgot Apple ID or password."
2. Enter your Apple ID. Did you forget your Apple ID?
3. Select the option to reset your password, then choose Continue.
4. Choose how to reset your password:
   • To answer your security questions, select "Answer security questions" and follow the rest of the steps.
   • To get email instead, select "Get an email." To reset your password, use the email that we send to your primary or rescue email address. Didn't get the email?
   • If asked for a Recovery Key, use the steps for two-factor authentication or two-step verification instead.

If you forgot your Apple ID password - Apple Support

Is the message an update account settings or verify account settings?

If you are using IOS 12, I read a post a couple of days ago where someone posted they had talked to Apple Support about this and were told it is a known bug that Apple is trying to correct. If you click Cancel, does the window go away? You can also try signing out of iCloud and back in.