Why does Mirror Entity copy Flightmaster Dungar including the chosen quest?

Based on the wiki page for Mirror Entity (Advanced rulebook — Playing/summoning a minion may also be helpful context):

• Minions are copied in every detail, including current stats, card text, and enchantments.

• As Mirror Entity minions are not played from the hand, any Battlecries possessed by them will not be activated. However, like all secrets, Mirror Entity takes effect after any Battlecries from the triggering minion.

So, the key here is that Mirror Entity doesn't trigger the minion's Battlecry, but it does copy the modifications already made by the Battlecry to that particular minion.

You should see the same results for any Battlecry minion which buffs or transforms itself.