Yet it does. Almost every single business/government service in America uses DoB + last 4 SSN to identify you. The two together make fraud trivial.
Before you answer, you may want to poke your answers into this site and have a look at the outcome: https://www.ssn-check.org/lookup/
Caveat: This tracks your issue date, not truly your birthdate. In the past couple of decades many/most babies get registered at birth, but if I stick my own (birth) data in there I actually get the wrong answer, because when I was born issuance wasn't automatic yet. But that will work for a lot of people.
https://www.ssa.gov/history/ssn/geocard.html
Anecdotally, while bored in math class we figured out that 10 or so of the guys in the class had one of two numbers for their first three.