attackfish

I assume Zuko didn't get into sword fighting until after his exile, a headcanon helped along by a number of fanfic writers who had written fic in which just such a thing happened. A thirteen year old Zuko, who didn't feel good enough, who didn't feel like his firebending was good enough, or who maybe even for a while was traumatized by his burning and was not dealing well with his own fire might have turned to swords to level the playing field in a way other benders wouldn't. Or Iroh may have introduced him to swords, as a way to keep him from constantly comparing himself to Azula. either way, like you, I doubt Ozai would have approved if he had known, and strongly suspect Zuko was away at sea when he started.