Sure. It's not a setting. Your microphone is a service of your computer and so it's relatively easy to grab a copy and record it. The Far Side is a service of Skype and Skype doesn't like to share. It's not entirely evil although it seems like that. Skype has to make sure its echo cancellation, processing and environment management works no matter what you did to your computer. They do that by taking over your audio, full stop.
Pamela is a software package that claims to be able to gracefully record both sides of a Skype call and split the conversations left to right so you can filter and correct one without the other. Mix down to a final show in Audacity whenever you wish.http://www.pamela.biz/en/http://www.pamela.biz/en/products/
Pamela now claims full Windows 10 support. For a while there, they were having troubles with Win10, and their support services got shaky, but historically, they've had the best track record.
You can mess with VoiceMeeter if you want. They claim to be able to make new sound pathways inside the machine and do custom sound routing as needed. Jury's out on how well it works. Post back if you decide to go that way. Remember, this isn't a help desk.http://vb-audio.pagesperso-orange.fr/Vo ... banana.htm
I didn't do it that way. I used two computers and a small mixer.
The computer on the left plays music into the podcast and records the mixed show. As long as you don't mess with it, the two directions on any computer are completely independent. The computer on the right is Skype. Skype thinks it has perfect control of its computer and doesn't know anything about the rest of that stuff.
That's how some of the larger podcasts do it.
Even if you do use fancy software on one machine, that may only work until Skype decides to change something or do an update. Several people have complained on the forum about their show turning to trash one day for no good reason.