They just have form-fitted hard jaws that bite into it. Could be a 2-jaw, 3-jaw, or even 4-jaw hydraulic chuck. Once the part is chucked, the fixture may even be able to index around to machine different sides/angles. Sometimes you can see the jaw/teeth marks on the un-machined areas. There are workholding companies that can come up with jaws or fixtures to hold onto almost any shape. This machining center fixture might give you an idea (it is
tough to find pictures of this stuff!) Picture a small version of this mounted on a lathe. You can machine one side, index it 90° without unclamping, and then do the other side/sides.
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