@Lars I Nielsen
Unfortunately, at this time FME Server does not have an inplace upgrade path that allows you to leave the existing FME Server software in place on the same system and upgrade that software. We do not provide patches or hotfixes either. This is something that users have asked for - review this idea (link to the idea).
That said, the Backup Tool will bring across all Automations/Schedules/Workspaces/Publisher/Subscriber/Users/Queues/Security settings (more details here). We do not bring across job log history. We do not restore any SSL/HTTPS configuration at this time. The licensing is not backup either. So you'll need to relicense the new system.
As for licensing goes, as david_r points out, the same serial number can be used to license the new server. If you need a transition period (new system) that will be fine too. If the system is online, it is pretty easy to get your server licensed.
Contact us if you have any issues.
Hope this helps.