You need to run two instances of OBS at the same time. Which is more complicated, as it hits your computer harder, because you're encoding everything twice. And you might have problems hooking the game twice, if you use Game Capture.
A lot of forum posts on how to do this can be found using this search:
https://obsproject.com/forum/search/164317/?q=multi target
The OBS re-write will probably make this easier, by the way, but its still in production.
So you're streaming music eh? I've already had one false positive on one of my VODs. I know its a false positive, because I don't add any music at all, and the game I stream (Planetside 2) doesn't use any outside music. No word yet on my petition to have it unmuted.