He does, that's not in question. Some of us are just baffled that he would give his name as "Duke of Sussex HRH" on a US birth certificate and not even something like "Henry Duke of Sussex" let alone Mountbatten-Windsor which some of his titled relatives have used on their own marriage certificates in the UK.
That's neither how his name or title is used in the UK, on Archie's birth certificate or his own when he was born.
He has four personal names and didn't include any of them. His choice but it seems a baffling one.
I used to be a death certificate clerk in Calif. The state regulations for death certificates clearly prohibit the use of titles. If you use a title such as Captain, Colonel, Father, Sister, Monsignor, Esquire, etc, it is not allowed to be used. Given names only, maiden name for mother or spouse. If your title is “Captain,” for example, that would go under your occupation. I would think the protocol about titles would be the same for birth certificates. Puzzling!