no, there's only a joystick interface on the gamecard, you'd need a seperate midi card like the PC-midi musicquest clone or similar, or just an additional soundcard and just use the MPU interface on that.
maybe there is a pin header for a joystick/mpu port or a midi out on the motherboard?

I didn't see one, but I may just not know what to look for.

well what motherboard is it?

I don't have the exact model number, but it's for a compaq deskpro en p550 minitower. Here are some pictures of the board and the part stickers.

From what I could find, there is no gameport / MIDI port on the Deskpro EN series.

Sooooo.. here is a crazy idea, or maybe a not so crazy idea.

On the Deskpro, use SoftMPU with the serial port.
http://bjt42.github.io/softmpu/

Then on the MIDI emulator machine, use Hairless MIDI <-> serial and loopMIDI:
https://projectgus.github.io/hairless-midiserial/

You will most likely need a null modem cable (serial crossover cable) to hook them together like this .

I think it would be easier to buy a new soundcard with the midi out joystick port. So another es1869 (is there a difference between the 1869F and the 1869FC? haven't found anything anything) unless there are compatibility issues I'm not aware of.

Look what is going to be available for sale in about a week. An RS-232 to MIDI adapter that is compatible with SOFTMPU:

MPU-232 : RS-232 MIDI interface, compatible with SoftMPU