What a “Prince” actually is
With few exceptions (and we’ll get to these in a bit), a British prince is a male whose parent or a grandparent is or was the King or Queen of the United Kingdom. Prior to 1714, the title of “prince” was not in customary usage except with regard to the Prince of Wales, the title a monarch could bestow on his or eldest son of the reigning monarch.
You can’t become a prince by marrying a princess
A man does not become a prince by marrying a princess. That is why Jack Brooksbank did not receive the title “Prince Jack” upon marrying Princess Eugenie. Similarly, a woman cannot use the title of “princess” just because she married a prince. The closest title she can use is the British title equivalent of a “Mrs John Smith.” For example, one of Meghan Markle’s official titles is “Princess Henry of Wales.” The only way you can use the title of “princess” before your own name is if you were born with royal blood, like Princess Eugenie or her sister, Princess Beatrice.
Incidentally, the woman we all think of as “Princess Diana” should never have been called by this title – her official title was “Her Royal Highness, The Princess of Wales.” When she and Charles divorced, she lost the “Her Royal Highness” portion of the title but was still entitled to be referred to as “Diana, Princess of Wales.”
You could become a prince by marrying a queen
Twice in British history, a man who not born into the British royal family became a prince, nevertheless. The first was Prince Albert, who after marrying Queen Victoria, was named by the Queen as her Prince Consort. The second was Prince Philip, who received his title from his wife, Queen Elizabeth II in 1957 when Elizabeth decreed he be made a Prince of the United Kingdom (he has been born a Greek prince, but gave up his Greek titles upon marriage). Both Prince Albert and Prince Philip were/are entitled to be addressed as “His Royal Highness.”