> Similarly, the ratio between the proton mass and the electron mass is approximately 1836

https://en.wikipedia.org/wiki/Proton-to-electron_mass_ratio says it is

  1836.152673426(32)

So, it differs about 0.153 from its nearest integer. Over 30% of all reals are that close or closer to their nearest integer, so I wouldn’t call that extraordinary at all.

For the fine structure constant, that’s about

  1/137.035999177

Closer, but still not extraordinary. The math is different but I think it’s about a 1:15 chance that a random real will be relatively that close to the reciprocal of an integer.