Hi AndersKrarupJensen,
Thanks for the post in Apple Support Communities. I see you may be running into an issue logging in to the App Store on a Mac.
To clarify, it sounds like you're signing in and it seems to accept your username and password, but you still see the "Sign In" option in the lower left of the App Store instead of your username; is that correct?
Does it help if you click Apple menu () > Restart? If not, try booting into safe mode to test. This runs a diagnostic/repair, and then prevents certain items from loading, so it will help to narrow things down: Use safe mode to isolate issues with your Mac. Once you reach your desktop in safe mode, try signing in, and then click Apple menu () > Restart to get back to normal.
I hope this helps get you started.
Cheers.