It is a quirk of history and actually fits with a modern view of a married woman keeping her own surname and passing it to her children lol.
Partially it is based on the strong personal views of three people on the Queen's accession -
Queen Mary, Queen Elizabeth the Queen Mother and Winston Churchill
And in part was a reaction to the supposed comments of Prince Philip's uncle Lord Mountbatten about the house of mountbatten ruling.
George V and George VI were both highly regarded personally and both their widows were loathe to see the house name change. Though it is alleged to be Queen Mary who spoke to a lady in waiting about it whose son was Churchill's private secretary who raised it with the PM who in turn raised it and advised the Queen to act.
An additional problem was both Queen dowagers were not overly fond of Lord Mountbatten.
The arguements to keep Windsor was helped by the confusion over Philip's surname and which house he belonged to - which hadn't been completely overcome by his adoption of Mountbatten when he became a naturalised British subject (which he hadn't legally needed to do anyway)
Philip's surname was essentially made up and was based on the anglicised version of his mother's name Battenburg rather than his father's (which would have essentially been the House of Oldenburg - or as a male line descendant of Christian IX - Schleswig Holstein Sonderburg Glucksburg itself a branch of the house of Oldenburg)
Under normal practice on the Queen's death - her heir would be the first ruler of the House of Oldenburg - or if Philip's invented surname was chose the first ruler of the House of Mountbatten - neither was acceptable to a lot of people in the 1950s.
So came the first decree not withstanding the Queen's marriage her family and descendants would remain the House of Windsor.
It was later amended, largely because Philip was deeply hurt by the decree, that those of their descendants who required a surname would use Mountbatten-Windsor.
Technically of course their children and male line grandchildren would all be Royal Highness and don't require a surname legally.
Prince Andrew and Princess Anne - both used it when signing the marriage registry but Charles and Edward didn't
Most grandchildren use their father's territorial designation as a kind of surname - William and Harry Wales (before William became Duke of Cambridge), Beatrice and Eugenie both use York, James would be styled Lord Severn (a courtesy title as the son of the Earl of Wessex) and Lady Louise is usually Lady Louise Windsor (though both Louise and her brother are still entitled to use HRH Prince or Princess)
There is one thing that always bothered me: Victoria was the last British monarch from the House of Hanover, and when her eldest son succeeded her as Edward VII, he took on the house of his father Albert, the House of Saxe-Coburg and Gotha, since it's traditionally always the house of the father that continues the line.
Now, the current British monarch, Elizabeth II, is once again a female, of the House of Windsor (which got its name after being rebranded from House of Saxe-Coburg and Gotha during the Great War); however, her descendants are members of the House of Windsor too, and not members of the House of Glücksburg, as their father Philip.
Can someone please explain this to me?