You'd need to use VAC/VB-Audio/Voicemeeter and set up a virtual audio system. Which goes well outside the scope of OBS support. Though I'm not sure if a browser can be pointed toward a custom audio device anyway, so you may be SOL and have to listen to the YT audio.
Skype can be split out though, if you want to ignore your stream in favor of talking with someone else. Not a good idea.
Just as a reminder, getting music from YouTube does not grant you rights to rebroadcast it, and is copyright infringement. Even if Moobot/Nightbot/Ankhbot/Deepbot/etc have music requests, it's still illegal to play copyrighted music on a broadcast without a license.