What is the name of this cable hanging between the sockets now?

Do both orange cables need to be connected to the router now? Or why are there 2 cables?

I still don't really understand the wiring at your place. Where do the orange cables lead to? But a cable that is not connected anywhere on one side is pointless, so it probably won't hurt to connect it to the Fritzbox as well.

I understand that the orange cables run from the utility room to the hallway. There are two connection points in each socket in the hallway. If I plug a patch cable into the left socket in the hallway, I would also need to plug the connection in the corresponding room into the left socket. I suspect that the second orange cable also needs to be connected to the Fritzbox. But what is the name of this intermediate piece that I still need to get? The electrician is never available, and I am no longer inclined to keep calling about it.

Get yourself a switch. This takes over the function of distribution. With a patch cable (your "Affenschaukelkabel") you go from the router socket to the switch and from the switch you can supply the rooms with a patch cable each. The slots for distribution are called "ports". You can get such a switch for about €20, e.g. with 8 ports from TP-Link.

There is a cable running from there to the hallway.

I haven't seen the other forum posts.... The switch has already been suggested

It could also be that only one Cat(the orange) leads to the double socket. Just give it a try. ;-)