Does it make sense, and are there physical reasons to think about the fine structure constant as a (very concentrated) probability distribution rather than a single real number?
The same question applies to other physical constants like the mass of a proton or the plank constant if you wish.
The question is not if these constant are not constants but varies with time or from one place of the universe to another (that people sometime discuss and was asked here several times); the question is if these constants are really constants, but they are described not by a single real number but by a probability distribution.
We can also ask if the fine structure constant is a random variable, does it have interesting physical consequences? Could it have consequences for computing quantities where we use it for computation, such as QED computations (say, for energy levels of the Hydrogen atom). Perhaps the effect on such calculation will be more interesting than simple averaging over this random variable.
A related mathematical fact is that in critical lattice systems of statistical mechanics, quantities (like the size of the connected cluster containing the origin) often have distributions which are "smeared" - the variance behave like square root of the expectation. (So the fluctuations do not vanish as $n$ goes to infinity.) I am not aware of such quantities that are directly related to fundamental physical constants though.
This (speculative) question arose from a discussion on my blog as a sharp alternative to the following (speculative, as well) possibility: Quantum computers will allow the computation of more and more digits of the fine structure constant, just like we can compute with a digital computer more and more digits of e and π.
Related question: QCD and QED with unlimited computational power - how precise are they going to be? ; ON TCS-stackexchenge Quantum-algorithms-for-qed-computations-related-to-the-fine-structure-constants;