The proposed solution via Grub showed no effect (I tried it meanwhile)
Using the brightness-control app, it is possible to control the brightness for the built-in and a second display
The brightness control via the keyboard (FN-F4/F5 keys) was not achieved
The solution is a workaraound.
For me it is sufficient since I don't tinker a lot with brightness.
@zabadabadoo When I asked for It I definitely could not see the "solution" marker in the elipsis menue. Now I see it prominemtly placed on every post. Guess it was because just having joined the forum hours before and thus not beeing granted rights to mark posts as solution?