COMING OUT IN APRIL, 2023:
and the sequel, "Hail Mary" https://www.amzn.com/1684334888
Available on Amazon.
Is it Windsor? Sussex? Markle? Does he even have one? Do any of the British royals have one? What name do they put on their drivers' licenses? How do they sign their credit cards?
Of course they probably have someone to drive them anywhere they'd like to go and buy them anything they need, so it's likely they don't need those sorts of nuisances in their lives.
So my guess is that Harry probably doesn't have a last name per se. I'm guessing it might be like what I used to tell my piano students:
Still, if he and Meghan are intending to stay here in the States, as it appears they are at least for a good while, he might find that he'll need to write something other than "Prince Harry" on the forms the IRS will surely expedite to him in order to collect their share of the multi-millions he'll hoover up from his newly-released book.