Update: According to the video mentioned below, even if an iPhone 13 display is replaced with a genuine and original iPhone 13 screen, Face ID will stop working. In the video, the repair provider swapped two original iPhone 13 screens and in both cases, Face ID was inoperative after the new screens were installed. It seems to be the case that if the original screen is put back with the original iPhone 13, Face ID returns, ruling out the possibility of an improper installation.
Apple has long faced criticism for making it difficult for third-party repair stores and providers to fix and repair their products. Apple already displays a prompt for customers inside Settings that informs them that their display is a "non-genuine display," and a similar prompt for "non-genuine cameras" was introduced earlier this year.
Apple claims these prompts help inform customers that they aren't using genuine Apple parts and is part of its more comprehensive plan to ensure customer devices are only repaired and diagnosed by certified Apple technicians.