I’m familiar with harmonics. Regularly spaced lines are indeed the expected result for a constant tone with overtones, and if the fundamental frequency varied then you would see all of the lines swooping up and down together. However, we’re not just seeing lines. Having harmonics swooping up and down at the same time, so that each harmonic almost reaches the frequency of the next band before returning, looks much more unusual.
It could be two tones with slow vibrato, played so that the vibrato is exactly 180 degrees out of phase, but that seems unlikely, and I can’t hear anything like that. Also, the shape of the curves looks more like half-circles than sinusoids, and you’d expect something more like a sine wave for natural vibrato or sine/tri for synthetic. It could be sidebands from something like ring modulation or FM, but I haven’t seen that sort of pattern come out of those processes.
Not just the regularity, but the similarity in energy from low to to high frequency seemed unusual to me. Now that we know the lower frequencies were cropped out, that doesn’t seem so striking, but it’s still quite top-heavy for acoustic instruments, speech, or typical synthesis techniques. However, my background is more in analogue synthesis, Chowning-type FM and the like, and I haven’t used some of the newer DSP-based softsynths that might do stranger things to the upper harmonics.
There’s also the possibility that the high frequency “circles” might be artefacts of sample rate reduction, bitcrushing, aliasing or MP3 encoding. But there’s still something odd to me about the fact that the range of the frequency sweeps matches the gap between harmonics, and the repeating echo-like patterns even early in the video when there’s no prominent echo on Elizabeth’s voice. They could indeed be coincidental artefacts, but the patterns look very different from all of your examples and I wouldn’t like to rule out the possibility that they were deliberately placed as visual patterns in a spectrogram.