The story behind is a bunch of trade-offs.
Manufacturing costs, winding costs, benefit or detriment of requiring a bobbin, size, stray magnetic signals, Eddie current losses at the required frequency, current density, coupling efficiency. Also, the various designs can be executed in multiple geometries so the "black material" is not limited to the square shape or E-core design. Once you start looking, there are many innovative designs to achieve cost effective coil winding, mounting and controlling EMI.