ThinkPad Universal USB-C Dock: Somehow recognized, but monitors, ethernet and fwupd don't work with T570

I have a "ThinkPad Universal USB-C Dock" (Link to German Lenovo page ) that works reasonably well with a DELL 5560: Monitors via Displayport, Ethernet, USB, fwupd all work. The DELL is running Debian 12 (Bookworm). I didn't have to install anything, just plug an d play. Now I wanted to attach the ThinkPad Docking Station to my ThinkPad T570 and was quite astonished that almost nothing works: No monitors are recognized at all, Ethernet says "no link", fwupd doesn't list the device. The t570 is also running Debian, only 13/Trixie. I had a look at the bis settings for thunderbolt: It is enabled, Security is set to "No Security". Please note: it's NOT the "ThinkPad Hybrid USB-C Dock mit USB A" station, where there are already some posts here. And I don't think it's related to DisplayLink, because I didn't have to install anything related on the DELL (also nothing related was already installed). How can I bring the ThinkPad Dock to work with the T570?