I have a couple thoughts. Not sure if they're helpful, but here goes.
FIRST:
Almost as a thought experiment, I'd analyze the problem purely from a musical perspective. It may be that what can be gleaned from a musical analysis is utterly useless in providing any kind of medical insight, but in the remote chance that it might, here goes.
If this were purely a musical example, two culprits come to mind.
The first is the phenomenon of when a sound cancels itself out at a certain frequency when it's out of phase with itself. For example, if you're playing music in a room with parallel walls, certain frequencies may reflect exactly 180 degrees out of phase, which cancels out that frequency. Think of, for example, a sine wave that is simultaneously encountering its mirror opposite. At any point in time, the sum of both waves equals zero. Thus, if the solid line in the following image was a musical tone, and the dotted line was its own reflection bouncing off a wall, the reflection would null out the original tone (at any point on the X-axis the value of the solid line added to the value of the dotted line equals zero), and it would seemingly disappear!
View attachment 37154
The second culprit that comes to mind is a lack of overtones.
Overtones, which complement fundamental tones, infuse musical sounds with richness and timbre. When you play a note on an acoustic instrument, you not only hear the fundamental pitch, but also a series of overtones, notes that you also hear but to a lesser extent, stacked up on top of it, that help give it its particular sound.
However, if you have a basic sound wave, a fundamental pitch without overtones, it tends to sound synthy and artificial. The "8 bit video game music" which you describe as the "only thing that sounds normal," is typically made from pure sine waves and square waves, without overtones.
SECOND:
I'd do an experiment where you take a song where you're clearly missing a part, and do two things: 1) change the pitch while keeping the speed constant; 2) change the speed while keeping the pitch constant.
The first, where you keep the speed constant but change the pitch could indicate problems at specific frequencies if your perception of the song changed. For example, if you raise the pitch, listen to it, raise the pitch again, listen to it, and do this a few times (or do the opposite, lowering the pitch), if you discover that the instrument you were missing reappeared and another instrument disappeared -- the distortion you experience changing as you change the pitch -- then that suggests there's damage at some frequencies but not others.
Alternatively, if when changing the pitch you don't experience any other changes other than hearing the song at a different pitch (the distortion remains the same, and what you couldn't hear before you still can't hear and in exactly the same way), then that suggests, I suppose, that the problem is more universal.
What that might mean medically is beyond me, but it might mean something to you or someone else with medical expertise.
Then, I'd also listen to the song keeping the pitch constant but changing the speed. It'd be interesting to see if the way you heard the song changed, if missing parts reappeared, for example, if it were slower, indicating your brain is processing music differently at different speeds.
There are a number of ways to change the speed and pitch of a song independently through software, but one of the easiest ways to do it is with Anytune:
https://anytune.us
Again, not sure if any of this is useful, but figured I'd ramble on a bit in case it were.
If you have any questions, I'd be happy to help.