I have an Apple ID that I use for iCloud, and another that I use for the App Store (and everything else). My attempts to login with my main ID failed, but it succeeded using my Cloud ID. I then logged out of the App Store, and was able to login using my main ID. I did get prompted a couple of times, but the second login with my main ID worked. At some stage I also had to authorise the computer.
Pretty certain I got the login credentials correct always because I’m using a password manager. At this stage I’ve not had any updates, so can’t be 100% sure the App Store is fully functional.