Apple's response, by way of MacRumors:
An Apple spokeswoman commented on the issue, referring to protective security features intended to prevent "malicious" third-party components from potentially compromising a user's iPhone as the main reason for the "error 53" message.
We protect fingerprint data using a secure enclave, which is uniquely paired to the touch ID sensor. When iPhone is serviced by an authorised Apple service provider or Apple retail store for changes that affect the touch ID sensor, the pairing is re-validated. This check ensures the device and the iOS features related to touch ID remain secure. Without this unique pairing, a malicious touch ID sensor could be substituted, thereby gaining access to the secure enclave. When iOS detects that the pairing fails, touch ID, including Apple Pay, is disabled so the device remains secure.”
She adds: “When an iPhone is serviced by an unauthorized repair provider, faulty screens or other invalid components that affect the touch ID sensor could cause the check to fail if the pairing cannot be validated. With a subsequent update or restore, additional security checks result in an ‘error 53’ being displayed If a customer encounters an unrecoverable error 53, we recommend contacting Apple support.