Oh man, I don't find it annoying, i find it super immersive! That and the turbo whizz you SOMETIMES get when downshifting (maybe at excessive revs) in the Porsche gt1 car... still can't work out exactly what situation causes it (or is it just random?) and it makes the car feel so alive.
I did some testing and I think it might be completely random in AMS2. But it might also follow some logic regarding different amounts of pneumatic pressure required by shifts at different amount of revs and gearing. And of course the same compressor system might provide pressure for some other systems in the car too. (Of which I wouldn't personally have any idea.)
So if I remember/understand how it works in real life, there is a finite canister of gas, that gets refilled by the compressor when it reaches some low level... and the compressor system monitors this (this is perhaps replicated in the video you posted). If this is correct, I suppose that the ideal system would replicate the "using-up". If different gears use different amounts of pressure, I suppose it might appear random after some cycles of up and down shifts. I'm trying to think of a circuit which would help to clarify...
Looks like it is the sound of a pneumatic pump that makes sure there's enough compressed air for gear shifts.