Alright, Paul_B found a document that helped actually.
Turns out there is nothing to do on my system. It's not like I needed drivers or so.
In fact the device itself has to ways of connecting. A(nother) manual stated:
To enter the bootloader, hold down the BOOT/BOOTSEL button
(highlighted in red above), and while continuing to hold it (don't let
go!), press and release the reset button (highlighted in blue above).
Continue to hold the BOOT/BOOTSEL button until the RPI-RP2 drive
appears!
So in case anyone has that problem, this is the (easy) solution actually!