Why won't Meghan Markle be called "Princess Meghan"?
To put it simply, the only way to have one's first name preceded by "Princess" is to be born into the royal family. So, for instance, Queen Elizabeth's sister Margaret was allowed to call herself Princess Margaret, and the Queen's daughter Anne is Princess Anne (and, subsequently, her daughters are officially Princess Beatrice and Princess Eugenie).
This is not to say Meghan Markle won't be a princess, she just won't be called Princess Meghan, with Princess in front of Meghan. Officially, she will take the title of Her Royal Highness, Princess Henry of Wales. That follows the example of her soon-to-be sister-in-law Catherine, who officially became Her Royal Highness, Princess William of Wales the day she married Prince William.
What will Meghan Markle's official title be?
It is rumored that as a wedding gift, the Queen will actually bestow Meghan and Harry dukedom like she did for Kate and William, which would actually change Meghan's title to Duchess (Kate's official title since she and William were bestowed the Duke and Duchess of Cambridge is Her Royal Highness, Duchess of Cambridge). Duchess is less a demotion from "princess" than a way to distinguish one's role in the Royal Family.
Dukedoms are often doled out to members of the royal family who occupy prestigious roles that are somewhat outside the main business of the monarchy, according a report by the BBC. If the rumors are true, the Queen will bestow them the dukedom of Sussex, and Harry and Meghan will thereafter be known as the Duke and Duchess of Sussex, with Meghan's official title becoming HRH, Duchess of Sussex. That doesn't grant them any land or control over any territory in Sussex or anything, but rather just a distinct and fancy title.