That's basically a simple oscilloscope in xy mode (turned by 45°, so y is left and x is right. You might say the shape is actually a diamond rather than a circle...). When you supply this with a mono signal on either channel, the oscillations make just a thin line in that channel's direction: the other channel is "dead" so the osci is degenerate to one dimension. If you pan the mono signal anywhere, it'll still give a thin line pointing in a specific direction (vertical for a center signal), because the channels are perfectly correlated. So for simple panned mono signal, the goniometer does indeed make a pretty good measurement device for the pan angle.
It gets a bit more complicated when you supply an actual stereo signal. Then, the channels are to some degree independent, hence you get not a single line but a whole tangle of arcs. But if the signal behaves mostly like a simple panned mono (e.g. because it is the output of a mixing console with such pan pots, or a stereo recording done with the xy-pattern), then you should still see an elongated shape pointing in the predominant pan direction.
For other stereo-mic techniques, there's another thing to consider: they rely more on timing differences rather than simple intensity. So just comparing the amplitude wouldn't tell you much about the stereo angle. However, these timing differences can still be seen on an xy-osci; the shape will be an ellipse. Deducing an exact angle from that is difficult, even with a tool like that goniometer; but at least you'll see there are phase differences.