I’m aware of how ANC works. It uses the same frequencies with reverse amplitude. The comment you objected to however was talking about different frequencies.
Right? One of my dumb noobie questions has always been why textbooks explain ANC as "180 phase shift" instead of "reverse amplitude". Do they really do an entire DSP pass to break apart a signal and phase-shift it just to get the output that a minus sign could get? There must be something missing from the plum-pudding model.
It's only a sign flip when the detector and emitter are perfectly in phase. Adding delay causes the required phase shift will drift slightly. Also note that the phase offset of a delay is frequency dependent, so this drift will be as well.
Another reason is that devices and materials have different frequency responses, so an external signal needs to be filtered to match the levels in the device.