tldr: If you need shared clipboard don’t run VMs in headless-mode.
During the last days at work I notices my shared clipboard in Virtualbox was not working. After checking everything I found this in the logs of the affected VM:
ClipConstructX11: X11 DISPLAY variable not set -- disabling shared clipboard
Why is there no DISPLAY set?
Well, the VM was started in headless-mode, everything makes sense now.