I'm not sure if you can listen to the music from the same output that your game comes through without it being picked up by your Desktop Audio Source in OBS. The only way I could think of is to change the output device of the specific application that your music plays through, although that would require you to have separate speakers or headphones.
To change individual application audio outputs, right click on the volume icon on the Windows Taskbar. Select "Open Sound Settings". Scroll down until you see "Advanced sound options" and click on "App volume and device preferences." There you can select whatever progam you are using, and change the Output device to any audio device that is not currently being used by OBS.
Alternatively, if you wanted to listen to the music and play through the stream at the same time, open OBS, and add a new audio source using the same device that you selected as the output device for your music. Then, right click on the little gear icon in your OBS audio mixer, and select "Advanced audio properties." Find the source that your music is on, and select "monitor only." That will only allow your music to play through your headphones AND your stream. If you wanted to mute the music in your headphones but still output to your stream, you can go to the same Advanced properties window and select "monitor off."
Sorry that I don't have a solution to your problem, but hopefully that can help you understand what you can do with your audio sources.
If anyone else has a real solution, I would like to know as well!