If I recall the books correctly, that was Geralt's decision...He expected that people will respect him more if he's authentic...But wasn't Vesemir the one who suggested to take the "of Riva" surname in the first place? He even suggested faking a Rivian accent.
It's the same name, the Name Geralt/Gerald (female: Geraldine) is Old high German and is composed of the words "Ger" (Spear) and "waltan" (to prevail). The old High German letter "t" had the same pronunciation as the Old English letter "d".Why do people call him Gerald instead of Geralt? Is it Gerald in other languages?