I did some more flights now. I have flown two airframes so far, the FedEx N692FE and the DHL D-AEAF. The problem only occured in the D-AEAF while flying in Europe.
Most of the times it happens, when the framerate of the simulator significantly drops. For me this is e.g. the case when I exit VR.
I have captured the problem now in a video. It looks horrible in the beggining, as I am in VR, but right when I exit and the simulator has its moment, the cabin altitude shoots to 69K ft.