Biden’s Press Secretary said this about the vaccines:

“…even though many of them [the mRNA vaccines] are approved and have gone through the gold standard of the FDA approval process.”
Press Briefing by Press Secretary Jen Psaki, 16 July 2021

She knows the vaccines aren’t approved:

The Pfizer-BioNTech COVID-19 vaccine has not been approved or licensed by the U.S. Food and Drug Administration (FDA), but has been authorized for emergency use by FDA under an Emergency Use Authorization (EUA).
Pfizer website

The Moderna COVID‑19 Vaccine has not been approved or licensed by the US Food and Drug Administration (FDA), but has been authorized for emergency use by FDA, under an Emergency Use Authorization (EUA), to prevent Coronavirus Disease 2019 (COVID‑19) for use in individuals 18 years of age and older. There is no FDA-approved vaccine to prevent COVID‑19.
Moderna website

In fact, the FDA isn’t even going to be considering approval until NEXT YEAR:
U.S. FDA Sets January Target To Decide On Approval Of Pfizer’s Covid-19 Shot

Lying erodes trust.