MTG Mirage Mirror copying Medomai the Ageless
"[Cardname] can’t attack during extra turns" places an attacking restriction on the card. 


  508.1c The active player checks each creature he or she controls to see whether it’s affected by any restrictions (effects that say a
  creature can’t attack, or that it can’t attack unless some condition
  is met). If any restrictions are being disobeyed, the declaration of
  attackers is illegal.


Therefore, any creature with that restriction can not be declared as an attacker during extra turns. It does not matter how the extra turn was created.

Note that Medomai has two separate abilities. They happen to work together by design (or rather against each other), but they are not connected ruleswise. 

Now the restriction only applies while declaring attackers. If Mirage Mirror was somehow attacking before it turned into Medomai (e.g. March of the Machines), that player would continue to get extra turns from Medomai's first ability. Similarly, it will also work if Medomai or a copy thereof is put onto the battlefield tapped and attacking (e.g. Ilharg, the Raze-Boar).