It is a pretty simple concept: If a kingdom makes a colony or conquers a country, they can either consider it to be a new chunk of the kingdom or they can organize it as an independent country, with the king taking over the role of monarch there as well.
For example the British could colonize Australia and eventually establish Australia as a country independently of Britain. Australia is not part of Britain, it is its own thing, but the head monarch is the same person.
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