One possible reason I can think of is that when Mirror Move was introduced in Gen. 1, the Pokémon that could learn this move were the Pidgey and Spearow line, both Normal/Flying types. And as these Pokémon are Bird-Like, this could explain the Flying-typing of Mirror Move.
Also, another reason is that some Birds can mimic sounds they hear, and while this is limited to sounds, not actions, it could explain why Mirror Move is a Flying Type move as well. I believe it is a combination of the 2 reasons I provided above as well as that Game Freak wanted Mirror Move to be a Flying Type move.
