It was an HDMI connectivity issue.
If there is a connectivity issue to an external monitor, for example HDMI cable is detached or damaged, the DeX Station behaves exactly as described above. Everything in DeX Station seems to be working except of the external monitor which remains blank and also there is no DeX item among notifications.
My assumption that the external monitor will start working after the DeX notification is clicked was wrong. The monitor should always produce the image from the phone when the phone is on. Only then the DEX notification follows on the phone.