However, if he is the King of England, isn’t he automatically the King of the United Kingdom?
The error you are making is that there is no King or Queen of England as such.
The Scottish and English crowns merged in 1603.
In any case the British Monarchy is seperate from the rest. The current Queen is also the Queen of Canada, Australia, New Zealand etc. It is their Monarchy too and it is seperate from the British Monarchy.
In other words the UK could become a Republic tomorrow and the Canadian Monarchy would not change one iota.
In 1801, the separate title of King of Ireland was merged into a combined title of King of the United Kingdom and Ireland for George III, and in 1922, when Ireland was partitioned, George V became King of the United Kingdom of Great Britain and Northern Ireland, and also separately King of Ireland (again). Precisely when Ireland ceased to be a Kingdom is complicated - but today the British Monarch is King of the United Kingdom of Great Britain and Northern Ireland.
He is not the King of England. Such a title no longer exists.