I think inverse square laws are so common because they apply to situations where the distance of one object from another (a one dimensional line) is used to calculate the force potentially felt at any point across the surface of a sphere (a TWO dimensional surface) at that distance.

But then you also have the strength of magnetic fields that follows an inverse CUBE relationship. The simple way I model this in my head is that magnetic dipole fields kind of fill a three dimensional volume with curved field lines, as opposed to gravity or electric charge where the “lines” go straight out, and at any specific distance the total strength of the omnidirectional field is spread throughout that two dimensional surface of a sphere of the same radius.