The government itself will remain aware of the character's legal name regardless of protestations. However, provided they are not attempting to set up a false identity for nefarious purposes, there is paperwork that can be filled out to set up a new legal name that would be used for business registrations, rental agreements, etc. as well as socially. It is similar to the real-life process of changing one's legal name, but no court hearings or public notifications would be necessary.