"Pigeon milk" does exist: it is a type of crop milk, the regurgitated lining that parent birds feed to their young.

But "pigeon milker" as pigeon fancier instead is a copy of the Dutch word for nightjar ("geitenmelker"), according to NL Wikipedia:

https://nl.wikipedia.org/wiki/Duivenmelk

Nightjars were known once as goatsuckers in English because it was believed they sucked milk from goats.

So it seems it is just a play on words. I am not Dutch, though.