I can't speak to all cases of amended birth certificates in New York City (and I've never been a lawyer), but I know the case of my paternal grandfather (even though I forget some of the details of what I saw).
He officially changed his name in the 1940s; I was told (when I was growing up -- probably in the 1960s) that the change was done to make life (specifically college admissions) easier for his sons (born in the early 1930s) -- so perhaps "antisemitism" was a factor; on the other hand, I don't think that he (who was a prosperous accountant when he got his legal name change) was in any more danger (in New York City -- or in the United States in general) in the 1940s than his 2 older brothers (both of whom had immigrated to the US as children; 1 older brother had already died, and his younger siblings were all women who changed their family name upon marriage) who were also still alive in 1940 -- and who kept their family name.
His original birth certificate (he was born in the Brownsville neighborhood of Brooklyn in early 1906 -- the first member of his household born in the US) had his name as "Joseph Kantor" -- with his date and place of birth (I think also the names of his parents as they were reported at the time) -- as it was written in 1906 ; when I saw the certificate on microfilm at the New York City Municipal Archives (a few years ago), I was pleased (but a bit surprised (and I think more than 1 Archives worker was surprised) to see (I believe) stamped notations indicating that his name was legally changed to "Jay Joseph Kantor" (early censuses give his name as "Jacob"; perhaps his "Hebrew name" was Yaakov Yosef (?) ) in 1940 -- but that the NY City Health Department amended his name as of a much later date (after World War II; I think about 1949 (my father has an Ancestry.com record concerning his Social Security Administration records stating that he was still named "Kantor" in June of 1948).
(I have confirmed -- via Newspapers.com -- that an official legal notice concerning the name change (by a court in Brooklyn, as I recall) to "Jay Joseph Kent" was published in the Brooklyn Eagle in July of 1940; I'm not totally sure why my grandfather seemingly didn't (seemingly) make public use of the name change for years -- although I believe that he became estranged from and then divorced from my grandmother before he publicly used his new name (and definitely was still married to her in 1940).
(By 1949, 1 of his sons -- I think -- was already at college; his other 2 sons (my father and his twin brother) were still in high school; I don't think that danger from "antisemitism" in either New York City or the US in general (he may have already traveled and bought property in New Hampshire) had *increased* from 1940.) )
So: I can say with confidence that legal name changes could result in New York City government amending birth certificates (not changing what was originally written, but including a statement concerning the changed name) to reflect the name change selected.
Ethan W. Kent in New York City
(researching my Grandpa Joe's Kantors (I pretty-much know the identifies of all the few Kents who resulted from the name change) -- as well as the 3 other main branches of my family tree (immigrant heads of household with last names of Paat/Pat/Patt/Pate (and possibly a non-permanent arrival record for the father in 1888 as "Pott"), Gelperin/Halperin, and Kornhauser.)