It's not unusual for someone to change their name when immigrating. Have you met anyone in your presumably English-speaking country who wrote their name in Korean characters? Probably not because they convert it to the nearest allowed name.

They don’t change names when they move. The names are registered in the location of birth.

But your name might get mangled in some states if it doesn’t fit, same as it might in the US if some system doesn’t properly handle, e.g., multiple middle names or special punctuation.

Dunno, but Iceland also enforces it's son-of/daughter-of naming convention to the point some names are explicitly prohibited (or is it explicitly allowed, I forget if it's a black-list or white-list).