The PSX only had D-pad style controls at release in 1995 -- no joysticks! The Dual Analog controller wasn't released until 1997.

(I think you're getting distracted by the terms "digital" and "analog". It may help to think about this in terms of discrete and continuous inputs instead.)

True, but there's a rather radical difference between an input device with nine possible states like a d-pad and one with 65,536 possible states like the N64 control stick.