In order to avoid the possibility of their signature being forged and used against them, royals are prohibited from signing autographs for fans. Yes, they may sign visitors books, official documents, and condolence letters, but they can't just give their signature to anyone on the street.
Back in 2010, it was royal rule-breaker Prince Charles who broke protocol to sign an autograph for victims of the Cornwall floods. While meeting with a couple, he was asked for an autograph for their young son that would "make his day." Charles then shocked his bodyguards when he asked one of them to find a piece of paper for him to sign. He wrote "Charles 2010" and even apologised for the "shaky writing" because he "never writes standing up."