Summoners can mount flying mounts and the Egi gets "stored."

Hell, if you have your chocobo companion out and you get the Midgardsormr mount via the questline, you appear mounted on Midgardsormr after the cutscene despite having your chocobo summoned. When you land, the chocobo appears, similar to how the Egi is stored and reappears when you dismount.

Clearly this is possible, yet I can't understand why we aren't allowed to mount anything we want while the chocobo companion is out.