You're on the right track. A MIDI port is not the same as a joystick port, but a joystick port can be implemented in such a way that, with the proper adapter for cabling, it can be used as a MIDI port. This is how the Sound Blaster was designed and most consumner soundcards follow this model.
It might be possible that an onboard sound chip can leverage a game port as a MIDI interface, but if your device had that capability a MIDI In and Out port should be present in the list of devices. I have a laptop at work with a on-board sound chip and it appears that there is no MIDI interface.
I suppose it's also possible that there might be a third-party software app that allows a generic game port to be used as a MIDI interface...