In Volume Mixer I can tell the green bar shows the loudness, but what does the grey bar mean?
What is the significance of it being clipped by the user-set volume level?
The grey bar indicates what level the application is outputting - i.e, what level the green bar would be if your volume was set to the maximum. It isn't clipped so much as scaled downwards.
The grey bar shows the volume of the audio that the application is sending to the audio-interface of the output-device.
If the output device has a volume-controller, the volume may be adjusted manually(volume slider) or even automatically (loudness equalization), meaning it can be lowered or in rare cases even increased (amped).
The green bar shows the volume of the audio that is simply being outputted, or in other words, being sent by the output-device to the audio reproducer.
Note that if the grey bar, and consequently the green at 100%, are not reaching the maximum HEIGHT it means that the application is not transferring the audio at maximum volume. A great number of applications nowadays have their own volume-controller so most of the times it's on purpose, for example to prevent audio-peaking.