The King of a thing, group, or organization, called "the red lions" has no reference to any other game. More importantly though, we know he is the King of Hyrule, and that is important, especially to timelines, as a royal bloodline with a male ruler has not always existed in every game.
As he was hiding out in
tWW, I'm guessing someone finding his true identity would be confused by the title "king of red lions" and no one would think him an important person. So it was a title to waylay those looking for him.