Think of it this way - identity theft works like a puzzle, made up of pieces of personal information. You don’t want to give identity thieves the pieces they need to finish the picture. One of those pieces is your date of birth. For example, just by knowing your date and place of birth, scammers sometimes can guess most of the digits of your Social Security number. Once identity thieves have the pieces they need, they can use the information to open new accounts in your name, claim your tax refund for themselves, and engage in other identity theft.
In general, it's not a good idea to share personal information online. Other examples of documents that can contain sensitive information are boarding passes, paychecks, credit cards, birth certificates, health insurance, medical records, drivers license, and even work emails. Always consider whether what you are sharing could have privacy/identity theft implications.
For more information, please visit the FTC consumer site:
https://www.consumer.ftc.gov/blog/2021/02/social-media-no-place-covid-19-vaccination-cards.