My dad had his name changed "illegally" as a child

Basically in the 1980s when my dad was a kid his mom remarried and instead of going through the legal ways of changing his last name, they just somehow got him a new passport with the new name. He doesn't have his birth certificate or proof of a name change. How can he prove now that he changed his name? Does he legally have to change it or sign a paper saying it was changed?

We tried requesting a copy of his birth certificate from his birth place but they wanted proof that he's the same person.

Location: ID/NH/VT?

Author: WrenTronic