I just learned about the Modified Discrete Cosine Transform (MDCT) and I see that it's not only used in mp3 compression, but is also often applied on overlapped power-of-two windows like an FFT. When I say "learned about" I don't mean I understand it, I just mean "became aware of".

I've always wondered why some of my FFT resynthesis patches, when run under smaller block sizes, have that low bitrate mp3 sound. I would describe it as squeggy bubbling. Are the two transforms related? Is that crappy sound due to something they both share, or is it just coincidence?

short answer: because mpeg conversion also includes FFT at one point.

That’s an interesting point.
The bubbling sound might happen because both FFT and MDCT work with overlapping windows and process data in a similar way. This could be why it sounds like low-bitrate MP3s