Yes, starting with version 7.2.1 for iOS, the private key for your Threema ID can be transferred to a new iPhone or iPad using Quick Start, which allows your Threema data to be restored automatically.
This setting is enabled by default and can be disabled under “Settings > Privacy > Easy Device Switch.” If this option is disabled, the ID must be transferred to the new device using the ID export or Threema Safe.
The Threema ID can also be transferred to a new iOS device via an iTunes/Finder backup. To do this, the “Easy Device Switch” option must be enabled and the backup must be encrypted. Apple does not have access to the private key stored in the encrypted iTunes/Finder backup.
The Threema private key cannot be saved in an iCloud backup. If you want to restore your Threema data to a new device via an iCloud backup, you should also enable Threema Safe.