I was finally able to solve the problem. I'd be happy to tell you how I finally managed to do it.
We have several WLAN networks and some of them were set up with VLAN tagging and not as bridged, but tunneled to a concentrator. We had to set this up so that all devices received the correct IP addresses. We briefly deleted this configuration and then undid it. Since then, we have also been able to establish a VPN connection. Everything works as it should.
However, the VPN connection does not work on the smartphone. No matter whether with Cisco AnyConnect or Android's own VPN solution. I'm still in the dark at the moment.