It's the wired Logitech M500 (latest Modell that has the shortened "logi" print).
I uninstalled the mouse, then restarted (press alt+F4 multiple times until a window comes up, then choose "restart" in the list and press enter) and it was back to normal (same 2006 driver again), thanks.
But: I now know what's causing this (kind of but no idea how to fix it): The mouse has got an extra button below the scroll wheel that switches between smooth scrolling (the way the wheel "runs"/feels in the mouse, not in actual Windows) and something that feels like it stops regularly (like a gear, this is the one I'm using). If I press that button (gear -> smooth), then clicking the scroll wheel goes back to opening that weird toggle modus. A PC restart doesn't help at that point, you have to uninstall the mouse again.
I tested the mouse on the 20H2 PC again and pressing the button there doesn't mess anything up. I also tested the 20H2 PC's M500 (slightly older model with the full "logitech" print) on the 21H2 PC and it's also not causing the problem.
What is that "toggle window" mode even? What keyboard shortcut usually enables it and how do I fully disable it?