According to Shazam Paper[1] (I can't find publication date), there were nothing fancy, just keypoints extraction on spectrogram. No deep learning with 100 layers or something like that which wouldn't be possible in 2000.
It's pretty obvious what GP meant. The concept of crowd-sourcing small chunks of work as a service is not the same thing as a hoax which involved a single person masquerading as an automaton. Amazon chose a clever name for its service that is memorable and references humans fronted by machines.
Ehh, I have seen references to using a person instead of an algorithm before Amazon released their service. Basically, if your automating human dexterity it's robotics, if your automating the brain it's AI. Sorting vegetables being a useful early example where humans could be thought of as a replaceable black box if you can break down what they are specifically doing. Thus Machine vision and classification where two common tasks because you need to play any game not simply replay a specific one.
The notion of a "mechanical turk" however, and of operations putting a facade on humans doing the work -- to which the parent alluded to ("a big mechanical turk sort of operation") and not specifically Amazon's variety , has been known and used for centuries.
[1] https://www.ee.columbia.edu/~dpwe/papers/Wang03-shazam.pdf