No wonder that the customer was not happy with this username to administer his Azure tenant. It created some headache and many discussions with many People at MS.
And dont believe that everybody there is on the same level and topic a subject matter expert. It may take some time to reach out to the right people.
Finally I found the right distribution list to ask excactly for this topic. And guess what. It is definitelly possible to move your Azure admin account to an organizational Account (local AD synced with Azure Active Directory).