There is a reason for this, a palace insider explained: In the royal family, self-promotion is considered a cardinal sin. Unlike the Scandinavian royals, who happily pose for photographs when walking into charity events, the British royals tend to stride by with their heads down, which can make them seem aloof and awkward.

Ms. Markle is not like that. She connects. In January, when a 10-year-old girl on a rope line asked her for an autograph, she grabbed the paper, wrote “Hi, Kaitlin,” and drew a smiley face. This circumvented an unwritten rule of the royal family, for a reason that dates back centuries: If a royal signature is made public, it can be forged. Her most sharp-eyed monarchist critics, mostly women who caucus anonymously on social media, noted it down. A few weeks later, she did it again, hugging a 10-year-old girl. (Tradition dictates a handshake or “small curtsy.”)

In this, Ms. Markle is similar to Prince Harry’s mother. Diana was congenitally unable to maintain a formal distance, something that endeared her to ordinary people — “commoners,” as they are sometimes called here — but that drove her palace handlers crazy. She was insistently human, and this stripped the royal family of some of its sheen. Particularly damaging was a television interview she gave in 1995, describing her private struggles with bulimia, depression and her husband’s infidelity.

Diana’s sons have made it clear that they plan to follow her model. Harry, in particular, has stepped forward as an advocate of modernization, telling Newsweek last year, “Even if I was king, I would do my own shopping.” This plan worries conservative royalists, as did the emotional interviews the princes gave on the 20th anniversary of their mother’s death.

“Each time you do one, you’re slightly peeling back the layers of the monarchy,” said the television presenter Piers Morgan, who has logged time on a series of Fleet Street tabloids. This, he is convinced, will be the way of the future: As the younger generation rises in prominence, “you’re going to see a more Diana-esque royal family,” he said, more “touchy-feely and emotional and heart-on-sleeve.”