Clearly if, as the pure textbook physics answer states, u is independent of surface area the "coefficient of friction is the same for all objects made of the same material.".
Actually, the coefficient of friction isn't a property of just one material, but rather is a property of the interface between TWO materials.
That is, you cannot say "the coefficient of rubber is .90" What you COULD say though is "the coefficient of static friction between rubber and dry concrete is .90"
The coefficient of static friction between rubber and asphalt is .85
or
the coefficient of kinetic friction between rubber and ice (sliding, not static - after you lose control) is .15
Note: there are no units on the coefficient of friction.
Skoorb has been correct all along in that generally, the frictional force is INDEPENDENT of the surface area of contact. Any difference in "grip" of the tires is due to other considerations (see my post earlier in this thread) such as wider tires being able to "take advantage" of more imperfections in the road's surface.
Also, F=uF is for relatively smooth surfaces. In the case of tires, you have the rubber being pushed into the small nooks and crannies in the road's surface, so in addition to frictional forces, you have to consider sheering forces on the tire. The entire issue is incredibly complicated beyond the level of basic physics. If fact, if it weren't, tire companies really wouldn't need R&D departments. But, back to the physics level,
the coefficient of friction DOES NOT matter on the surface area of contact. BUT, the interaction between road surface and tires isn't one strictly of just friction.