The way the term is used is as if it is divorced from the fact he is the king. Were not all kings princes? Did his rank as king of England not take precedence? What is the relationship between the two titles?
The word was used in the same way we use the word "monarch" or "sovereign (n)" today, as a sort of generic, neutral alternative term for king or queen. The definition of the term at that time was not as rigid as most people today believe it is.
In Elizabeth I's letters, for instance, particularly to Mary of Scotland, she refers to herself as a prince.