I think it does. Whatever our choice of units (provided you use the same choice of units in all calculations), the ratio of mass of the proton to the electron will be the same. This is what makes the constant fundamental: it is independent of any way we choose to measure it, so should be the same to anybody and everybody. If we were to meet an alien civilization and study their physics, as long as we know their number system, we would immediately recognize the number 1836.2.