It happened again! In a different town.
This time, I nearly watched it live how it happened: the city built a new road. The new crossing for this road was inserted approx. one normal sized building before the bus stop. Afterwards this bus stop was not serviced, the line took the new road instead and routed back to the bus stop on the opposite side.
The line fixed its layout when I deleted the bus stop from the timetable, but got weird again when I inserted it again. I had to replace the bus stop again, then it was accepted – but now the bus stop on the other side got broken! I had to replace this one as well, now everything is fine again.
The story continued and now I am not sure, that the new road built by the town was the trigger. I noticed that other lines in this town had this problem as well. Then, when fixing one of those issues, another one occurred (on a different line). This continued a few times, now everything is solved.