There is some connection between the King of Red Lions and the Fishman/men, as the first Fishman Link meets tells KoRL that he's repayed his debt. I don't think this has anything to do with the "Red Lions" as a group. It's probably just a cool name, or a take on the red Phoenix under the Triforce on the Hylian shield. Both lions and phoenixes have noble connotations associated with their use, so it makes him all the kinglier.