The brutal answer is that, by the standards of British royal protocol, the former Hollywood star lacks “royal blood”. That means that she cannot call herself Princess Meghan. The arcane rules mean that Meghan will be compelled to follow the example of her future sister-in-law. When Catherine Middleton and Prince William were pronounced “man and wife” in 2011, she automatically became Her Royal Highness, Princess William of Wales.
The overwhelming likelihood is that the new royal fiancee will in turn officially become HRH Princess Henry of Wales. So we can forget about Princess Kate and we can forget about Princess Meg. The reason for this is simple: Catherine Middleton was not of royal blood, and what applies to her will also apply in the case of Meghan Markle.
Royal blood meant that the Queen’s late sister Margaret was entitled to call herself Princess Margaret. Likewise the Queen’s daughter is Princess Anne and her granddaughters are Princess Beatrice and Princess Eugenie.