Do you mean custom music like Spotify/USB drive as you are playing? likely copyright barriers. Say they did allow a 3rd app to allow custom music to be played. Someone will likely record a match with that particular music playing via Youtube and Youtube gives them a strike because it's music that's patent by an artist and playing in the background while not giving profitable returns or credits.
Free Step Dodge
DestructionBomb
DestructionBomb
If anything, I don't think it's because they don't want to, it's just that they can't when there are barriers involved.
Princeofhalcyon
But there are a lot of games that do let you, and the ps4 doesnt save that music in video clips, nor can people you're playing against hear it, you'd have to add it yourself. and if people are on pc they dont need the usb to play music during a game. So it still doesnt make sense to me