The word "prince" in English comes from Latin "princeps" (genitive principis) "first man, chief leader; ruler, sovereign," noun use of adjective meaning "that takes first," from primus "first" (see prime (adj.)) + root of capere "to take" (see capable).
So, a prince does not have to be the son of a king and doesn't even have to be of a monarchy. The Israelites had some form of princes even before they wanted a king in 1 Samuel 8.
They had princes already in Numbers. Numbers 7:2: "That the princes of Israel, heads of the house of their fathers, who were the princes of the tribes, and were over them that were numbered, offered:"
A prince is simply a head, the principal leader, similar to the word principal of a school, he is the first or highest authority figure in whatever context that is, and yes, usually political contexts at least in English.
In Numbers it seems that princes were the highest leaders or representatives from every tribe from among the families of the tribes.