But what is this institution, and how does it operate?
Sometimes referred to as "the Firm," the monarchy works like a public body or government department, albeit one that is completely independent.
At its head is the Queen, and below her comes a strict hierarchy based on the line of succession. At present that runs down from Prince Charles to Prince William, Duke of Cambridge; his children George, Charlotte and Louis; and then Prince Harry.
There are also thousands of people working behind the scenes to keep the royal household running smoothly. These employees are spread over five main departments, including the Private Secretary's Office, which oversees constitutional matters, and the Privy Purse and Treasurer's Office, which looks after finance, human resources and IT.
During the interview, Meghan herself drew a line between the members of the royal family and the household staff.
"So, there's the family, and then there's the people that are running the institution," she told Oprah. "Those are two separate things. And it's important to be able to compartmentalize that, because the Queen, for example, has always been wonderful to me."
In one instance, Meghan said, she appealed to a senior staff member and the palace's human resource department for support with her mental health. She was told that as a member of the royal family, rather than a paid employee of the institution, she wasn't eligible for the department's help.
While the senior royals aren't on the household payroll, the Queen leans on them to represent her because she can't do everything herself. In return, they can access public funds known as the sovereign grant, which is paid to the monarch for his or her role as head of state and head of nation.
Prior to their decision to "step down" the Sussexes were part of this group, but due to the hierarchy their role was essentially to support the Cambridges.
The hierarchy also meant their household office was smaller than that of the Cambridges or Prince Charles, offering less support.
But receiving any money from the sovereign grant means the royals have to be publicly accountable for public funds and can't make money from their name.
They have to remain politically and commercially independent, avoiding any conflict of interest.
Taking a royal position means there are compromises on personal freedom, as members of the family represent the crown, not just themselves.
In a bid to enjoy greater freedom, Harry and Meghan wanted to become minor royals, who have no access to taxpayers' money -- but their plans raised serious issues.
The couple had a parallel, independent office in the US, where the palace had no oversight over the projects they were working on. This created tension with the "institution" referred to by the pair, who ultimately decided to end their royal duties entirely.