Just bought a Surface Pro 4 and have the same problem. Music stops the instant the screen turns off, every time. (Using Spotify, only Music player I've used for years.) That the problem has been here for years without a fix worries me...
I tried the screen saver option mentioned above but the screen doesn't really turn off then and is a bit dark gray not entirely black... for now I guess it's the best that can be done? :/