Face ID is paired to the board, not to the screen
The Face ID sensor array sits in the notch or the island at the top of the screen assembly, but it is paired to your logic board at the factory. That pairing is what keeps the biometric secure: a sensor array from another phone will not authenticate on yours.
So a screen replacement does not break Face ID, provided the original sensor assembly is moved across to the new screen. That transfer is a normal part of the job and it is why an iPhone screen repair takes longer than the panel swap it looks like.
The one case where it cannot be recovered
If the sensor array itself is damaged, usually by the same impact that broke the screen or by liquid, Face ID is gone and nobody can restore it. Not us, not Apple, not anyone, because a replacement array cannot be paired to your board outside the factory.
We check the array during the diagnostic and tell you before starting. A phone without Face ID works normally in every other respect and you fall back to a passcode.
The parts notice in Settings
After a screen replacement, iOS may show a notice saying it cannot verify the display is genuine. This appears for any display Apple did not fit, including genuine Apple panels fitted by an independent shop.
It is a message, not a fault. Touch, brightness, the sensors and Face ID are unaffected by it.
True Tone
True Tone works from calibration data stored in the original display. We transfer that data across where the model allows it, and where it cannot be carried over we say so before you commit rather than after.
If True Tone matters to you, mention it when you drop the phone in.
Touch ID on the older models
On the iPhone 7, 8 and SE the fingerprint reader is in the home button, and the button is paired to the board the same way Face ID is. We transfer your original button to the new screen and Touch ID keeps working.
If the button is damaged, Touch ID cannot be restored. Same rule, different sensor.