Works perfectly fine for me... I just ran a test as it is not my habit to play music and game, but it worked first time with no effort.
I love my Hill Climb Racing and always play it mute so it is set up that way.
I opened Modern Music, started a shuffle. Hit the Windows key to get back to start screen. Opened Hill Climb Racing and completed a run on the Dessert level with my pimped out Monster Truck while I listened to Gladys Knight and the Pips sing "Midnight Train to Georgia."
No problems. It may be your music player or your settings within that player.
I know phone calls, alerts and Health and Fitness goals will all pause the music, but then it resumes when those are done.
I kind of suspected I may have tweaked something in settings, but I looked in my Modern Music and there are no settings to account for it.