First, why 3 phase instead of single phase? Two reasons. 1) It enables delivery of 1.73 times more power for a given amount of conductor, due to the increased "fill-in" of the current sine wave-form with the timing of the wave-forms. 2) It can provide automatically a rotating magnetic field for motors.

Second, why not the others you mention?
-- 2 phase power provides advantage 2) from above, but not 1). 2 phase at 180 degrees separation would be indistinguishable from single phase in it's effects for motors. 2 phase at 90 degrees separation is commonly used to start motors on single phase systems, with the second phase generated at approximately 90 degrees with the (usually) temporary connection of capacitors at the motor.

-- 4 phase power provides advantage 2) from above, but not 1). 4 phase at 90 degrees separation would be indistinguishable from 2 phase at 90 degrees separation, see -- 2 phase above.

-- 6 phase at 120 degrees separation would be indistinguishable from current standard 3 phase. (so why do it?). 6 phase at 60 degrees separation would provide advantage 1) and 2) from above and run motors designed specially for it slightly more smoothly, but at the added cost of 6 contactor poles and isolator switch contacts and overload protectors. Not worth it.