Also just because you don’t recognise someone as your Prince, doesn’t mean you have a Prince of your own. Personally, I’m British so if I told the Queen of Denmark you’re not my Queen, you could assume oh he was talking about the Queen of England being his Queen.
But to the Americans on this site, if Prince Charles went over to the US, and started telling people in the street what to do, if one of them turned around and said remember you’re not our prince, that wouldn’t mean that they had their own Prince, it just means what it says on the tin, he’s not their/your prince.