Though the Red Hood Gang’s story later involved Jason Todd and then got retconned, part of Red Hood’s lore is in the mistaken identity aspect, whether it’s Joker not realizing he’s been set up as a hood-wearing patsy or if it’s a gang that changes up the “leader” position to throw authorities off or, in Gotham’s case, because they keep shooting each other.
Comic book fans know that in some iterations of The Joker’s pre-Clown Prince days, he took on the persona of the master criminal Red Hood.
Perhaps not incidentally, the Joker has been described as a possible ideal or state of mind that will affect several characters, which is kind of like the passing of the Red Hood mantle.