Is there a way I can share my saved flights with a FlightView app on another mobile device?

Yes! Sign up with FlightView and create a profile (tap Options button -> Settings -> Sign Up), and then your trips on the first device will be saved to this profile. Then log in with the same login and password on the second device and your trips will be downloaded to it. Go to the Trips tab and you will see them listed!

Please note: You must use the same login on both devices. Currently, it is not possible to share trips between different user profiles.