I think I am closer to the problem, by testing the following: I just renewed factory settings again. The MacBook Air asks my Apple ID, password and I can fill in the security numbers which appear on my iPhone. So far so good.
Then occurs something bizar, this happened two days ago as well, when I renewed my factory settings.
A second ID, for which is asked a password, appears. I recognize this mailadres as by trying to solve the problem, we tried a few days ago to make a new ID with this mail-address, but then we changed it again to the current ID (mail address). Even on the site appleid.apple.com I can only see my current ID.
How can I cancel this apparently second ID, asking for a password? Would this be the cause of the problem, not being able to login with my current ID??