There is an important point missing so far. The indistinguishability of a system (particle, atom, etc.) has statistical consequences. As fermions, protons and electrons follow Fermi-Dirac statistics. This means that systems consisting of multiple such identical subsystems have to be described using anti-symmetric wave functions.
If one is dealing with bosons, the wave function is symmetric since they follow the Bose-Einstein statistics, with the eponymous condensate being one example of the effect.
This still doesn't explain "why", but it shows that the indistinguishability is more than just a failure of measurement. I guess if one knows QFT and QCD one could give more insight, like
>>13808705 and
>>13806691 are implying.
