A nasty message as visible above suddenly popped up out of the blue. Every configuration change, stopping or starting the service was blocked because of that. The message was related to the Kerberization but "Disable Kerberos" option was also under the spell. It seemed that the only option was to plough under everything and build up the cluster from the bare ground.
The problem is described here but no solution is proposed.
Solution
The solution was quite simple. Remove the "Kerberos" marker from the cluster by modifying the Ambari database. In the case of Postgresql database execute the command:
update clusters set security_type='NONE'
After that magic, the "Enable Kerberos" button is active and after performing the "Kerberization" the cluster is happy and healthy again.