As this thread still gets updates from time to time, let me answer my own question:
The problem occurs, when there is more than one instance of the vSphere Client running at the same time. I do this frequently, as I have more than one vCenter Server to manage. Even if all vSphere Clients are closed, check that no failed or hanging processes "VpxClient.exe" are around. Kill them, if necessary. Restart a single new vSphere Client, and the remote media will connect.
Starting the vSphere Client as Administrator is not necessary. The fact that it is running under another user id is probably the reason this also works.