First of all,
friction is the force that arises between surfaces when there is relative motion between surfaces. And when the system is in equilibrium even after applying the force in some direction we say static friction is acting and this is the only adjusting friction
now as you have written coefficient of static friction =f/N (where f is the force and N is normal reaction). This equation is ONLY valid if the system is in equilibrium and force acts in the direction, in which there will be relative motion.
I also know that it depends on that surface area of contact and the material.
this is the biggest fallacy coefficient of friction never depends on the area of contact(think about this point and if confusion comment)
if N is halved then coefficient of friction should increase but that does not happen, it remains unchanged, why?
N can be halved only in one way in the picture i have shown and the way is by applying force in upward direction, so in this way N is reduced , but by doing this to keep system in equilibrium in horizontal direction force required will also be reduced (because in this $\mu N=friction$, N is reduced in)
Only static friction is self adjusting, right!