No you need just the one phone for a family all booked under the same flight reference number. You have to input all their details separately, Once you have checked the whole family in on the app, you just swipe across to see each individual boarding pass.
Also once the family are booked in on the one phone, then each
individual family member can also download the app and use same ref no to get the boarding passes automatically on to their own phones. This is fine for partners and teenagers. They can all just swipe their phones on the self scanning machines at the airport.
It's quite advisable to have a back up on another phone or on a tablet etc just in case you do lose, drop or break your phone. Once when I was going through departure gates, I switched on my phone only for it to start optimizing. Luckily I had a spare phone with the boarding pass backed up.