After close inspection, I noticed that the whirring noise the Cacodemon makes sounds strikingly similar to one ambient sound you would hear in Q3DM4 (among several other maps) near the pulsating tubes. The Cacodemon's noise sounds like a sped-up version with the last part of the noise reversed.

Recycled sound, or just similar?

id recycles themselves a lot since they were founded, ever since Wolf3D used Keen4 sounds, etc :P

and for the record all those games that go all "woah these sounds are from doom!" aren't directly copied from Doom. They sample from the Sound Ideas libraries, series 6000 in particular.

Edited 57 seconds after the original posting.

they also reused some atmospheric sounds from quake 2
I know cuz I have the steam version of quake 2 and it lacks the soundtrack :/