Yes because she's very fond of her father - but she shouldn't have done and she didn't need to - HRH The Princess Anne of the UK of Great Britain and Northern Ireland would have done or simply Anne
To give you an example Prince William's birth certificate gives no surname for himself, his father or his mother (only her maiden name Spencer)
they are described respectively as HRH Prince William Arthur Philip Louis, HRH Prince Charles Philip Arthur George Prince of Wales and HRH The Princess of Wales (maiden name Spencer) and the Prince of Wales signed the birth certificate just Charles.