Should I enter my Apple ID and password?

Only the core iOS system can request a login as such. Apps cannot on their own do that. They'd request the system to generate a login request if they need to such as when verifying in-app purchases or similar. And the login is processed directly by iOS and only a login account token is ever made available to an app. Never the actual credentials.

As long as the login request is a system popup and not a random form on a website or inside an app, you should be fine.

Nope. Apps can only generate screens when they are in the foreground. They cannot unilaterally activate a system popup like that if they are not the active App. Only the iOS core can request a login like that without an App running.

Also the likelihood that an App could get through Apple's verification process and onto the App Store with a login request function outside of the system functions is exceedingly slim. It's really difficult to imagine an App unilaterally requesting a login like that, and even more so when its not active.

It was very likely just an attempt to refresh a login token to something similar from the iOS core. Logging out and back in normally refreshes all login tokens so the request would have been satisfied.