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It does but not because it influences friction at the wheel contact point (the mass cancels out in that formula). If braking is done entirely in "wheels locked" fashion then all that matters is the friction coefficient between the wheel and the track. But most braking is not like that. A train will only lock the wheels in emergency braking as that will apply larger braking force than otherwise available.

In normal braking the friction between the pads and the wheel is the important one and in that case the stopping distance is determined by how much of the energy of the moving vehicle you can bleed through the force you apply with the braking pads. More mass/speed, more energy, more time needed to apply the xxxxN of force to the wheel and convert the energy to heat. The energy of the moving vehicle scales with its weight while the maximum force a friction braking system can apply doesn't.

The science of braking is even more complicated than that, materials heat up or melt, friction coefficients change, tires behave differently under different loads, ABS systems kick in, etc. These are deceptively complicated topics.

The formula for friction also doesn't contain surface area and yet we use wide tires and big brake pads. But the bottom line is that in a real life scenario (as in not in simplified formulas on paper) the weight of the vehicle very much influences the braking distance.