George V formally adopted a house name where one had only existed by convention before. In doing so, genealogically he and his descendants remained members of the Saxe-Coburg & Gotha branch of the House of Wettin, even if they will not personally use these names. When Elizabeth II declared that she and her descendants would retain the house name as Windsor, it did not change the fact that her male-line descendants also belong to the Glücksburg branch of the House of Oldenburg. They just won’t use that name.
Since Charles was declared to be a member of this House of Windsor, all of his legitimate male line descendants will also be members of this house unless he declares another name. He cannot unmake them as members of the Houses of Glücksburg and of Oldenburg though; it’s just genealogical fact even if it doesn’t come with dynastic jurisdiction.
The other present reigning hereditary monarchs of Europe belong to the following houses:
Belgium: House of Saxe-Coburg & Gotha, a branch of the House of Wettin; their official house name though has been ‘Belgium’.
Denmark: House of Glücksburg, a branch of the House of Oldenburg; Margrethe II’s heir belongs to the House of Monpezat.
Liechtenstein: House of Liechtenstein; the principality actually gets its name from the princely family.
Luxembourg: House of Nassau, a branch of the House of Bourbon-Parma, itself a branch of the Spanish House of Bourbon-Anjou, a branch of the House of France (the Capetian dynasty). The first House of Nassau died out in the male line and this is effective a second/new agnatic House of Nassau.
Monaco: House of Grimaldi, a branch of the House of Polignac. When Monaco passes through a female line, the house name is declared to be Grimaldi.
The Netherlands: House of Orange-Nassau, presently a branch of the House of Amsberg. Since the Netherlands has passed through the female line three times, the house name is declared to be Orange-Nassau each time. The three queens preceding the king were agnatic members of the Houses of (Orange-) Nassau, of Mecklenburg, and of Lippe. While the grand dukes of Luxembourg are truly the heirs to Nassau, the Dutch monarchs have a better claim to Orange than to Nassau despite hanging on to ‘Orange-Nassau’.
Norway: House of Glücksburg, a branch of the House of Oldenburg.
Spain: House of Bourbon-Anjou, although Bourbon is just used. This is a branch of the Capetian dynasty/House of France.
Sweden: House of Bernadotte.