[shakna]

In my construction, all the work is done by an alternative interpretation of "the".

The Crown example still works as the Parliament is not selected by the monarch. It is not elected by the monarch. The monarch does not possess the power to reject them. Yet, the Parliament are still referred to as "the Crown", and possesses the executive power of the Crown.

Equally so, the Governor General of any Commonwealth Nation is free to reject the orders of the monarch, as they possess the executive power of the Crown. The monarch is also free to fire them for doing so, as the monarch also has the power of the Crown. Both are on equal footing. In the words of Whitlam "Well may we say God save the Queen, because nothing will save the governor-general."

Multiple people have possession of the executive power in such systems, even though the word "the" is used to refer to it. That alone is not enough to indicate that a singular individual controls it.