UPDATE: I found it! I knew I had something that looked like that somewhere.
It appears to be the same part as pictured in the manual: each 3.5mm jack is in the same place, and the motherboard header is the same.
Comparison of the bracket side of the part I found against the manual:
Comparison of the motherboard side, showing the key and pin 1 are in the same position
Bracket side, including cable
Motherboard side, including cable
With the soldered wires into the external connectors exposed
Close-up of 3.5mm jack solder points
Close-up of one side of the game port solder points
Close-up of the other side of the game port solder points
Edit: Added photo comparing motherboard side to the provided manual
I thought I had something in my parts bin that looked like the bracket mentioned in the manual, but couldn't find it.
I couldn't find any photos of the part online, either, but it may look something like these:
This is a pretty common 15-pin Game/Joystick/MIDI port used on many motherboards from around that time
Single bracket with separate 25-pin parallel port and 9-pin serial port
Those kinds are the most common, with wires connecting headers on the motherboard to a simple port screwed into a bracket. There are no electronics have any kind in those examples.
However, it's also possible to have a circuit board connected to the bracket, instead of the wires going straight to the port, like this:
There isn't any "logic" on this board (i.e. no microchips), but it has some supporting components to (I'm guessing) help with keeping the signal clean, or provide protection to the motherboard electronics.
Anyway, if you do go the route of testing the pinouts, I'd guess the audio pins are the ones closer to the "Sound Pro" chip, simply because there are 15 pins at the "top" (relative to the orientation of the image in the manual) of the header including the key pin, and the joystick port uses 15 pins. This is purely a guess, but should be reasonably easy to verify (or at least rule out) with a multimeter by seeing if the +5V is available on pin 1 the pin next to the key (16). If that's the case, then depending on whether pin 2 or 14 is also +5V would say the orientation of the joystick pins (assuming the motherboard manufacturer wired it the easy/typical way which is not always the case).
You can see a typical 15-pin joystick header pinout here: https://www.frontx.com/pro/p0912_030.html
If that works, then you've narrowed down the pins to test to the 10 "below" the key. I'd also guess each of the 3.5mm jacks use three wires, each with L/R/Gnd, so maybe try to find which of those 10 pins have continuity with ground, and that might give some hints to where the left/right signal wires are for each. I'd only tap the probes very briefly on each pin if doing a continuity test, since technically the multimeter sends some power down a probe to see if it (and how much) returns on the other. But I don't think any damage would come from doing that kind of test, since they're connected to something directly exposed to the outside and are typically better-protected than headers designed for use completely within the chassis.
Anyway, sorry, it's not the answer you were looking for, and you may have thought about all these things already, but that's where I would start if I had that board.