I am in a constant signin/password loop when signing into my Account Settings in the App Store on my Mac Mini.

Have you tried restarting the computer? Are you running any anti - virus programs? Using a VPN?

Try setting up another admin user account to see if the same problem continues. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later. 

Isolating an issue by using another user account 

If the problem is still there, try booting into the Safe Mode using your normal account.  Disconnect all peripherals except those needed for the test. Shut down the computer and then power it back up after waiting 10 seconds. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. When you reboot normally, the initial reboot may be slower than normal. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart by using the application un-installer or by using the developer’s uninstall instructions. For each disable/delete, you will need to restart if you don't do them all at once. 

Safe Mode - About.    

You are welcome. The Safe Mode might help. All you can do it try.