I was having this issue going from 5s -> 6s but I was able to successfully fix this on iphone 6s last night, here's what I did.
1. Delete all passes/cards from Wallet (You will need to temporarily switch the phone to zoomed view to remove passes)
2. Under iCloud settings toggle the Wallet option to off so it is not storing data on the cloud
3. Remove the Wallet from app switcher by double-clicking home button and swiping up on the Wallet app
4. Reboot the phone
5. Create a NEW backup from the new phone (You don't want to use the one you created with iPhone 5/5c/5s)
6. Settings -> General -> Reset -> Erase All Content and Settings
7. Restore the phone from the NEW backup you created in step 5
8. Add passes / cards back to Wallet which should be displaying normally now
Hope it helps.