Sort of. Look into VB-Audio, and/or VAC (virtual audio cable).
That said, being on Skype and ignoring your audience, or letting your audience only hear half of a conversation is going to alienate them, and probably make people leave. No one likes being ignored, especially when it's a streamer's job to entertain their viewers.
Likewise, streaming music from Spotify is still a copyright violation. Even if Twitch mutes VODs with matching audio and takes no further action, that's to cover THEIR backside, legally speaking. You can still be sued for infringement directly, even if no one has been. Yet.