Birth certificates are routinely amended when a name is changed legally. This can happen with adoptions, petitions for name changes, etc.
However, it's important to understand that a person can use any name he or she wishes as long as it is not for fraudulent purposes. My father's birth certificate lists him as Louis Slominsky (misspelled, should have been Slonimsky) in 1917. The 1920 census lists him as Lawrence Slonimsky. Sometime around 1940 he and his brothers changed their last names to Sloan. My father enlisted in the Army with the name Sloan though one brother used the name Slonimsky.