Hager technical control center or server cabinet?

It depends on how many devices you want to connect. ;) Although I wouldn't necessarily equate the number of network outlets with the number of switch ports.

Do the Unifi APs also have the IGMPv3 problem or only the switches? And is there still no solution for it?

Does the combination of Cisco switch and Unifi APs work well regarding VLAN management and such?

Does it really have to be a 48-port switch then? €800 is quite a price.

We just have two 24-port patch panels that are almost fully occupied. I didn’t feel like having to worry about PoE when unplugging something. In other words, if I plug something PoE-capable in anywhere in the house, it should just work directly without me having to unplug anything else.
On top of that (and this was the biggest challenge) was finding such a switch that doesn’t mess up with Magenta TV.
After a lot of research, the Cisco was chosen because of these factors and other requirements.

Do the Unifi APs also have the IGMPv3 problem or only the switches?
And is there still no solution for it?

Does the combo of Cisco switch and Unifi APs work well regarding VLAN management and co?

The APs are not a problem. I have four running on the Cisco here and Magenta runs wonderfully.
The Unifi switches are the problem. There is also the Edge series from Ubiquiti, which can handle it too. I initially considered that as well, but there was just no experience available regarding it.
With the Unifi switches, there is a workaround, but it is quite complex and, if I remember correctly, has to be done again after every reboot.
The Cisco SB series is quite widespread and, for example, popular in the A d m i n I s t r a t o r forum.
Regarding VLANs, it can do all of that; among other reasons, that’s why Cisco was chosen. But honestly, everything is still running pretty “dumb” here because I simply haven’t had the time and inclination to tackle it yet.

I am running a Cisco SG 300 as a Layer 3 switch with 3 Unify access points and Magenta TV and 5 VLANs. No problems. The dumb Fritzbox provides internet and the switch routes between the individual networks and the internet as desired. In this way, I have also blocked my IoT network from the internet, but can still access both the internet and the IoT network from the other subnets. I did some reading beforehand and invested one of my wife's night shifts to implement it this way.

If you can live with the routing performance, that's okay. Otherwise, the SG300 is rather weak as a router.

For a small home network, more than sufficient and extremely stable. But it also depends on the data throughput and routing throughput one has. Most home users will not even come close to utilizing the performance of the SG 300 or its successor.