I found the cause of my problem at least, and have heard form others with the same experience.
Two of my SKR mini E3 V2.0 were bricked by an electrostatic discharge of myself. I just found out because I heard a clicking sound doing some calibration prints and wanting to remove filament residue, but not even touching anything - I just came near. This board then had the exact same issue like the ones before.
So if anyone has the same issue with the board as I had (blank screen, no serial connection/octoprint, can't read MC with STLink), it quite possibly came from an ESD and bricked the MCU/electronics.
So be sure to:
ground yourself before doing something on the printer, regardless where
use an esd-bracelet
don't keep the room humidity too low