My understanding -- from what an attorney who attends our local trans support group has said about the issue -- is that this would apply to people born in New York, and only them.
Birth certificates are kept/issued by the state where someone was born, regardless of where the person moves to, so whether one can change one's birth certificate depends upon where one was born. According to the attorney, the state where I was born (Virginia, USA) is fairly backward that way ("The land of the Boll Weevil / where the laws are medieval").