The column doesn’t yet have it quite right. The name “Mountbatten-Windsor” was declared to apply only to descendants (in the male line) of Elizabeth II who are distant enough that they no longer qualify to be called “His/Her Royal Highness”.
The reason for the change is that she is a woman, and, just as women change their names when they marry, so royal dynasties change when a queen regnant marries. There may be some technical question as to the surnames of the royal family, but George I through Victoria were of the House of Hanover, after which, through Victoria’s marriage to Albert, Edward VII and George V were House of Saxe-Coburg-Gotha. Then George officially changed both the surname and the house name to “Windsor” during WW1. At the same time, Prince Louis of Battenberg changed his name to “Mountbatten.”
So when Elizabeth married Prince Phillip, a descendant of Prince Louis, the House name would have changed to “House of Mountbatten” in the normal course. But Elizabeth felt that “Windsor” was a good name to stick with, and declared early in her reign that her descendants would continue to be known by that name. Later, because the Mountbatten family is also popular in Britain, she amended it to let her non-royal descendants use “Mountbatten-Windsor”.
So the present situation is that it’s “House of Windsor”, but the surname is “Mountbatten-Windsor”.