My computer is still showing remnants of the old Apple ID
Apple does have several places for id - I have for historic reasons 2 different ids and so note some of the differences.

There are actually separate places that have logins.


iCloud/Mail login
iTunes login for music/vides etc
iPad/iPhone app store
MacOS store
Apple developer (I have several different ids here)


I have one id for 1 and another for the rest and others for 5. But I think the Mac Appstore could be set as a different id.

2 and 3 might  be the same (although that could be historic as iOS apps were updated via iTunes)

This does mean that you will need to change id separately for 1,2,4 and 5 (I think 2 and 3 are covered by one change)