Background music path is stored in the save game files because you can swap out the background music with action parts/Lua script.
Also, I highly recommend never using the continue music from previous scene option because it will either play whatever music was playing in your menu/load/save scene after loading a save game or it won't play anything at all.
One thing you can & should do is create an action part in an at begin of scene action for each scene & set the backgroud music for the scene with the action part, this is just a fallback in case it wasn't specified or doesn't automatically play the scene background music that you specified via scene properties.
Quick note: you will likely encounter various random bugs after loading old save game files after making various changes to your project - especially if you have removed or added new things because the data in the save files will not contain the new/edited information.
In other words you should not use the save system while your game is in development. You should wipe the saves & start fresh each session - also it's a good idea to only use the autosave system while working on your game too as it's easy to check if an autosave exists via action parts & delete it in the game launch actions section.