I did not really solve it. I looked around for a bit but found no good solution. The only thing that sort of worked was to copy the stream URL in Jellyfin and paste it into VLC Media Player, which could play the videos okay. But that was not really satisfactory for me.
In the end I installed a new OS on my PC which already came with the codecs to decode x265 preinstalled. I installed it for other reasons but it fixed also that issue and now I can play any video in the Jellyfin Media Player. I did not try if it also works in the browser. The OS I am now using is called Nobara.
I do not think they are called drivers, but yes, installing the needed codecs should work. I just did not find out how to do it properly.