iPhone 16e setup fails to restore backup from iPhone 13 mini

Once the 16e is updated, you would erase the 16e and start over with a new setup. The erase will not revert the iOS, it will stay at 26.2.1

Settings > General > Transfer or Reset iPhone > Erase All Content and Settings

When you erase, you will have the option to keep existing eSIM or delete. Choose the option to keep the eSIM so you won't have to contact the carrier to start all over from scratch.

When you start the new setup of the 16e again, you will have the option to use the backup of the 13 mini to restore data back to the phone. That will take some time 45-60 minutes or so....depends on how much data will be transferred.

The most common reason that this issue occurs is that the 16e has an older version of iOS installed than the iOS on the iPhone 13.

For example, if the iPhone 13 has been updated to iOS 26.2.1 and the iPhone 16e has not, that would likely explain the difficulties.

What iOS version is installed on the iPhone 13? What iOS is installed on the iPhone 16e?

Good luck. Hopefully the storage capacity on the 16e is equal or greater than the storage on the 13 mini.....if the 13 mini was close to full storage.