While I'm unsure of what to call "butterfly magic" as a whole, you could have moves named after certain species of butterflies. For example, Morpho Shield or Monarch Rush.
Is he the only person who can control creatures like this? If that's the case, then most people would call him "the butterfly dude" or something unless there was a name that was propagated either by himself or the people who picked a fight with him and lost. Maybe the Butterfly Viceroy? Something with a species in the name, preferably.
Reminds me of a mix between Spider-Sense and the Scout Flies from Monster Hunter: World. Butterflies already swarm when there's danger, but that's usually because something has disturbed them from where they were chilling out beforehand. Give him a big cloak or loose trenchcoat so that a whole bunch of butterflies can hang out in the folds or the pockets. Something like Swarm Cloak sounds good to me, but I'm not entirely sure.
Your character reminds me of Agitha from The Legend of Zelda: Twilight Princess, mostly her Hyrule Warriors incarnation. Check her out, and you'll see what I mean. Bugs are her forte, but she primarily uses Butterflies and Beetles.