Hello,
I have finally made it working. I tried with a new room account and then to change it with "enable-mailbox room" but the same error happened. Then I found out somewhere that a command "Get-Mailbox room_alias | Set-Mailbox" could help and in deed, it helped. So something was not syncing somewhere (very vague specification, I know) and this command forced it. You can tell that this command did something when the output of the command is nothing. If you run it again it says "WARNING: The command completed successfully but no settings of 'account_name' have been modified." So obviously the previous run of the command did something. Sorry, I do not know what exactly.
After this I resynced the AzureAD and truly the online account was not enabled in Exchange online so when I logged with it into Teams it showed the on premise calendar of the room.
I do not know what is the exact cause but maybe it helps somebody.
Thank you for your help, it put me into the right direction.
Petr