But your assertion is not true. The FDA's site confirms the same fact that the CDC says:
https://www.fda.gov/news-events/press-announcements/fda-approves-first-covid-19-vaccine
"Today, the U.S. Food and Drug Administration approved the first COVID-19 vaccine. The vaccine has been known as the Pfizer-BioNTech COVID-19 Vaccine, and will now be marketed as Comirnaty (koe-mir’-na-tee), for the prevention of COVID-19 disease in individuals 16 years of age and older."
"While this and other vaccines have met the FDA’s rigorous, scientific standards for emergency use authorization, as the first FDA-approved COVID-19 vaccine, the public can be very confident that this vaccine meets the high standards for safety, effectiveness, and manufacturing quality the FDA requires of an approved product,” said Acting FDA Commissioner Janet Woodcock, M.D.