It’s because in Japanese, “Twister” is “Tatsumaki”. “Tatsu” roughly translates to “Dragon”, so Twister being a Dragon-type move is a pun.

Re: this.

Why do you think Hurricane’s a Flying attack but Twister a Dragon attack?

I suppose probably because Twister is in some sense a “magical” attack, in a similar way to Fairy Wind, while Hurricane is notionally about wind created purely by the force of one’s wings.  Why does that matter?  I suspect there’s probably not a satisfying answer.

Update: this.