itch.io

You could use the win-capture audio plugin for OBS, however it doesn't work reliably, and in some cases not at all, on newer versions of OBS. The only real way around this is to just capture desktop audio, unfortunately.

You can use softwares like Steelseries Sonar or Logitech Mixline. They split desktop audio into channels you can record. (I would recommend Mixline. It has a worse overview, but in my experience it is not as buggy and bloated with additional features you probably do not need.)

Just one important note. AVOID STREAMING copyrighted music. There are some artist such as PandaBeats specifically allowing use in streams, but use of copyright music can lead to strikes, muting of your VOD and other consequences.