Surely you'd agree that all electrons and protons are $exactly$ the same (indistinguishable).
Now consider a regular Hydrogen-1 isotope: A proton and an electron bound together. There's definitely more than just one of these atoms in the observable universe.
Well then consider an electron floating around the $n=2$ shell of a Hydrogen-1 isotope, then falling down to the $n=1$ shell (this event will emit a photon) - I think you'd agree this has probably happened more than just once in the history of the universe.
Every time this has happened, a photon was emitted, with a particular wavelength (determined by some equation). Different experiments might yield different measurements of that photon's wavelength (due to limitations of measurements, like you mentioned). However, according to Quantum Mechanics all the photons emitted through the aforementioned process had exactly the same wavelength.
Maybe there's some (unbeknownst to me) QFT corrections which will alter the wavelengths for different settings in the above scenario, but I think you get what I'm getting at. There are certain processes which occur in the universe that are perfectly reproducible, which will emit photons of the exact same wavelength.
EDIT: As many have pointed out, I completely overlooked broadening of the line widths due to the uncertainty principle, so my example doesn't work at all.