Looks like a pretty standard threadless headset to me, unless I'm missing something in the photos. That wedge you've found in the steerer tube is a star nut, used to tighten down the top cap and sandwich the whole headset stack together.
A diagram from the venerable Sheldon Brown:

Once that star nut is installed, it's more or less permanent unless it needs replacing. So, you're right: this is not really intended to have an adjustable stem height.

Edit: I see now, that stem doesn't seem to have any pinch bolts, so the top cap is also tightening against the stem to keep it in position.