What drives the shape of an engine's intake on a fighter jet?
Are the design drivers solely based on supersonic pressure loss improvement?


Yes. And sometimes a few more considerations need to be taken into account.

We have covered the reason for intake shapes well on this site, but the details must be collected from several answers. In a nutshell: Higher flight Mach numbers need more elaborate and heavier intakes in order to convert the kinetic energy of the flow into pressure. A pitot intake is light and simple, but a poor choice for speeds in excess of Mach 1.6.

Rectangular intakes are normally chosen in order to make internal, moveable ramps possible which create a cascade of shocks in order to slow down the flow as efficiently as possible. Such designs are typical for Mach 2 aircraft (think Concorde, F-14, F-15, XB-70). Another way would be a central intake with a moveable spike, but those had to make way for big radar arrays in more modern designs. The JAS-39 intake is a little special: It sits next to the fully moveable canards, and those need a vertical wall in order to minimise the gap between canard root and fuselage/intake structure. Yes, a round intake would be slightly lighter, but is impossible due to the situation of the canards.



Detail of JAS-39 canard, taken from this picture.