So I don’t forget:
If the drivers for an FTDI USB-serial adapter don’t install resulting in an "unknown device", try to remove all other USB connected devices.
In my case it was a PeakCAN USB adapter, connected via a rather long USB extension cable, that caused the problem (not sure whether it was the adapter itself or the extension cable, that caused it, but I assume the latter, because these adapters usually don’t cause any trouble).
Temporarily disconnecting the cable let me install those USB serial adapter drivers.
The problem was reproducible: It happened after each reboot.
Another unproductive hour spent troubleshooting a Windows driver problem. I just hate Windows and its opaque driver installation method. Give me real error messages Microsoft, so I at least get a clue what’s wrong!