A "prince", literally speaking, is not necessarily a "future king". He is the son of a king, and so MAY be a future king, but may not. If a king has two sons, they are both princes, but it is quite possible that one will become king and the other never will. In English the word "prince" is also used for the husband of a queen when it is the queen who holds the power. Like the husband of Queen Elizabeth of Britain is called "Prince" Phillip.
By extension, the word "prince" is often used, in the Bible and elsewhere, as a general term for a person with power or authority.
I am not aware of any more specific meaning in the Bible as the word is used in general. Of course one could examine specific uses of the word and discuss exactly what sort of power is being described, but then we're not really talking about the word "prince", but rather about a specific person who is called a prince.