The problem is that music isn't generally FOSS - people would like to be paid for their work. So the next best thing is to set up your own music service where you can add your own music.
That's true. However, FOSS usually means that it's free (F) and open-source (OS). Music isn't usually open source - they don't usually share the sheet music or allow free editing/covering/forking of the music. So the worlds don't completely mesh. You can look at a FOSS solution for managing music, but you won't find a ton of FOSS music.
I don't have much experience with FOSS alternatives for Spotify because I rarely use the service, but maybe you could find what you're looking for in this list from alternativeto.net ?!