Meghan Markle is not formally known as “Princess Meghan”. The Queen has instead granted Prince Harry the title of Duke of Sussex and as his wife, she is now known as The Duchess of Sussex.
Britain’s royal protocol is incredibly strict, and as such, because she was not born into the Royal family she is unable to be referred to as “Princess Meghan”.
The same rule applied when Catherine married Prince William. Her official title was “Her Royal Highness Princess William of Wales”, as well as “Her Royal Highness The Duchess of Cambridge.”
This means that Meghan Markle’s official title is “HRH Princess Henry of Wales” as well as “Her Royal Highness The Duchess of Sussex.”
Archaic rules from British peerage mean she takes the title of her husband. Because Meghan is not a “blood” royal it would be improper to call her Princess Meghan. She still technically ranks as a princess, but it would be improper to refer to her as Princess Meghan, and Catherine as Princess Catherine. Their status as princess comes from their husbands, and not by blood.
The Queen traditionally gifts her sons and grandsons Dukedoms in a friendly gesture of approval, as well as ensuring their wives do not have to go by their husband’s name. If this isn’t done, you end up with women going by their husband’s name, as we see with Princess Michael of Kent.
When Diana married Prince Charles, her official title was “Her Royal Highness The Princess of Wales.” However, the media had other ideas and incorrectly referred to her as Princess Diana. After the divorce, she was known as Diana, Princess of Wales.
Another odd case of this rule is in the case of Prince Philip. While he is from royal blood, being known as a prince of Greece and Denmark before his marriage, he was not officially known as a British prince upon his marriage to then Princess Elizabeth. It was only in 1957 when The Queen decreed that he should be “accorded him the style and title of a Prince of the United Kingdom.”
Technically, this means that she could decree that both Catherine and Meghan be made princesses. However, the traditions surrounding the issue mean it will most likely not happen.