It is all about the distribution of pressure under the contact. For a block of uniform weight the pressure can be assume almost constant under the area and so when traction is broken it will happen all at once all over with a force of $\mu N$ as you stated.
But for other geometries, or for elastic parts (like tires, or marbles or billiard balls on felt) the contact pressure has various other shapes. The parts with the highest pressure (sometimes in the middle, and often at the edges) are going to stick more than the unloaded parts. The result is complicated, but in the end we call the total traction to achieve full slipping still $\mu N$, but with $\mu$ a different value that for the block above, even if the materials are the same.
A lot of scientific papers are written on the subject of how traction affects the contact properties and vice versa. Dealing with a coefficient $\mu$ for the force and not the area makes it easier to summarize the results, but in reality the pressure shape over the area is the ultimately in control here.