No there is not far too many copyright issues, keep in mind that the soundtrack is sold on cd and is streaming platforms like spotify and prime music. She is also a legit artist even though it is all digital. Even discord now which had less of a problem with copyrighted material being streamed has now taken a more firmer stance on copyright material being streamed.

I record this game using Nvidia's capture software and the audio remains intact. The game will put a small watermark in the corner, but it's not intrusive. As far as I know, the PS4 game will mute music in recordings, but not livestreams.

Audio for Future Tone and DIVA X on PS4 will always be muted if you use the PS4/PS5's screen recording function. If you want to record videos with audio you could always just use PS4/PS5 screen recording and edit in the audio afterwards using Capcut or something like that. If you want to live stream the game with audio though, you'll need to get a capture card.

Although if you're streaming to friends on discord...you could just use your cellphone to livestream and record your TV. Audio and video quality would be very poor, but you wouldn't have to buy anything to do that.