From a cursory reading of succession and Houses in royal families it seems that tradition requires Houses to be inherited through patrilineange. And so when the Monarchy passes through a female there is a change of House.
Some examples of this:
- The House of Hanover was the House Queen Victoria belonged to, and her son Edward VII was a Saxe-Coburg-Gotha King, inheriting the house of his father. (Of course this name was changed to Windsor during WW1...but still remaining the same House)
- The House of Stuart changed to the House of Hanover as succession passed through Queen Anne
- The House of Tudors changed to the House of Stuart as succession passed through Queen Elizabeth I (Notably the House did not change between Queen Mary I and Queen Elizabeth I because they were sisters - coming from the same father)
- The same is true for the change between the House of Norman and the House of Plantagenet as succession passed through Empress Matilda. Though those terms exactly were not used at that time; instead it was "Norman Kings" to "Plantagenet Kings" but same idea.
The other times Houses changed in England were because of other reasons (i.e. Wars of the Roses).
So what gives? Why is Charles III not the first Mountbatten King?