There's your answer ne1atall.
The reason is that the Playstation System isn't a multi-tasking operating system. It's only designed to run one program at a time, which simplifies game programming, and makes game performance more predictable. Another reason is the memory model. The Playstation consoles make almost all of the system memory that isn't used by the System software available for games to use. There isn't any reserved for the music player to run at the same time as the game.
The XBox consoles, which are no doubt your reference point, run a minimal build of Microsoft Windows NT; which is a multi-tasking operating system, and uses a more complex memory model. In some ways it is more complex to program, and has less predictable performance, than the Playstation Systems; but Windows is the dominant PC gaming platform, so many programmers are familiar with the challenges of programming HD games on Windows. Sony isn't going to create a multi-tasking version of the PS3 or PSP System because it would not be compatible with the existing games for the consoles. In the case of the Vita, it just decided not to add that complexity to the software just to implement features like cross-game chat, and unrelated music playing during a game. Game developers don't care about those features, and many gamers don't either.