Since AMOffice© licenses are workstation licences and not user licenses, it is crucial how many computers/workstations are registered.
To solve the problem, open the ASA-AMOffice© Server Management Tool and switch to the tab 'Licensing'. On the lower left side you can see your licensed workstations.
After clicking 'Release workstations' and waiting for 10 minutes, your previous workstations will be reset.
You can then log on to any system with your license key.