Is there an app that allows an iPhone-to-iPhone local network call with shared music?
Short answer no. I found a link to an article that explains this.


  Playing music or any other audio while on a phone call is easy:
  
  While on an active phone call, tap the Home button to get to the home
  screen Open the Music app, find any song or podcast, and press play
  Return to the phone call screen by tapping the green titlebar Music
  (or any audio) plays immediately, but instead of outputting through
  the external speakers, it will be played through the ear speaker. This
  prevents the music from being piped through to the other end of the
  phone, and that means the other end will never know you’re jamming to
  Milli Vanilli to make it through their tortuously boring phone calls
  and wait times.