Just in case it helps anyone.
1. I've updated the Catalina to 10.15.1 and got the black screen on my RX 580 (cause I didn't update the WEG).
2. I booted from High Sierra backup from USB drive, updated the WEG in Clover/Others (got the black screen)
3. Updated the WEG to 1.3.4 in /L/E using sudo cp in terminal. I tried to rebuild the kext cache, but with no luck, cause I'm not sure how to do this when booted from the backup USB. (I still got the black screen).
4. Then I booted from recovery and reinstalled Catalina 10.15.1 (multiple reboots, choosing the Install from Clover).
That somehow worked for me - the video card model showed correctly, and the first impression, that everything works like it should.