JERUSALEM — At an Israeli children’s hospital, Melania Trump handed out blue canvas backpacks filled with Dr. Seuss books and emblazoned with a picture of the White House. At a forum in Saudi Arabia on combating extremism, Ivanka Trump stood in for her father when he got tired. And in Israel, Jared Kushner stuck by his father-in-law’s side as President Trump pushed for peace in the Middle East.

The president’s first foray overseas is giving the rest of the world its first real glimpse of America’s new governing structure — a White House that has quickly become an all-in-the-family business, standard practice in Saudi Arabia, but rare in the United States.

During the president’s nine-day, multicountry trip that began on Friday, the world is learning — and Americans are being reminded of — the significant overlap between blood relations and policy advisers in the current White House. For Mr. Trump, they are almost interchangeable.

“It is extremely rare that a president would bring family members, other than a spouse, and ask them to assume a public role on a foreign trip,” said Carl Sferrazza Anthony, the historian at the National First Ladies’ Library.