exactly, same thing for α: if it were different (numerically) that would mean that we are talking about a different universe, i.e. we would measuring a different fine structure constant. People in that universe would still call it "α", because we define "α" to be the fine structure constant. That wouldn't change the (numerical) value of α in this universe.
In the same way, we define PI to be "the ratio of a circle's circumference to its diameter", and that depends on the kind of geometry where you lay that circle on.