the Princess of Wales, who was formerly known as Kate Middleton, was indeed called "Catherine, Duchess of Cambridge" officially after her marriage to Prince William, Duke of Cambridge. However, many people still referred to her as "Kate" even after her official title change.
The use of "Kate" as a nickname might stem from her earlier years before becoming a member of the royal family. It was a common nickname she used, and the media and the public often referred to her as "Kate Middleton" before her marriage. When she became a part of the royal family, some people continued to call her "Kate" out of familiarity, habit, or simply because it's the name they were most familiar with.
In many cases, people use nicknames to create a sense of closeness or informality, even when referring to public figures or individuals with formal titles. The media also often uses more accessible or recognizable names to make information easily understandable to a broader audience.
It's essential to recognize that how someone is referred to can be a matter of personal preference. While her official title is "Catherine, Duchess of Cambridge," if she is comfortable being addressed as "Kate," it is respectful to honor her choice of name. However, it's always best to refer to individuals by their preferred name or title.