The MIDI port and the game port are two sepparate devices.
It's just that the Sound Blaster card used some unused pins of the DB-15 connector that was used by the IBM Game Control Adapter card.
And since Sound Blaster became so popular, almost all sound card makers continued to support this choice.
A normal game card has no MIDI pins, thus can't work with any of the usual MIDI devices.
https://mysite.du.edu/~etuttle/electron/elect57.htm
Edit: In other words, the gameport on a sound card can communicate with MIDI and a joystick the same time.
It just needs a splitter cable (MIDI/Joystick), in a similar fashion as it is required for a two player scenario (two joysticks on one DB-15).
Speaking under correction, I believe the easist would be to make your own 1:1 extender cable, with three wires (MIDI RX, MIDI TX, GND)
going to a separate connector (DIN directly or DB15) for MIDI.
Edit: You can also disable the game port on sound cards, independendly of the MIDI port (MPU-401).
Doing so would cause no conflict with a dedicated game card.
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