Backing up Amiibo data is as easy as using the Identify functionality from the top right on the main page.
Once Amy have detected your Amiibo, you will be presented with an option to backup your data.
On the Owned list, select the character you would want to restore and then tap Restore Data. You can then place your device on top of an empty NTAG215 NFC card/token and wait for it to complete.
DO NOT move your phone away from the NFC card/token until the checkmark has been displayed and the window closed.
Only NTAG215 NFC cards/token/stickers are supported. They will also need to be empty. Amy will do some checks before attempting to write you data.