Domain Admins can't connect to SQL Server
I found the answer to my problem in this answer by Remus Rusanu.

As an domain-admin connected to that SQL-Server-Host via Remote Desktop. I had to start SQL Server Management Studio explicitly "as an administrator". (right-click -> run as administrator). 

If SSMS isn't started in administrator mode, the current login is not in "builtin\administrators"-group. That was the problem.

It comes from Windows user access control (UAC). So that is actually a security "feature".