App Store popup keeps asking me to sign in every day

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Seems you might need to check that your Mac is properly signed in with your Apple ID (in System Preferences) so you can use Apple services ok any time you want, without reminders popping up..

Sign in with your Apple ID - Apple Support

Thank you for your reply, and for letting us know.

Re: Is there a way to find out what exactly is forcing the popup?

As the pop-up does seem to be App Store related:

You might want to reach out the App Store Support Team:

There is a "Get Support" link at the bottom of their web-page (link below)

App Store - Official Apple Support

or

if you want:

There's a webpage for sending feedback to Apple (link below).

They likely won't reply, yet your comments would reach those able to make changes via future updates.

Product Feedback - Apple

or

Apple phone numbers around the world

Contact Apple for support and service - Apple Support

That might do something. You also might want to restart the computer after signing out. Then sign back in once the computer has restarted.

You're welcome!

Thank you for your reply, and screenshot.