My opponent cast a Mana Dome and the cast the Arena within the area of effect. When I tried to use the arena to move my creature next to it, I got an error message saying that couldn't be done because of mana dome.

So it looks like at least two things are wrong:

1) Although the enchantment belongs to my opponent, the check was treating it as "enemy" since I was using the ability

2) It seems like the use of enchantment abilities should be unrestricted. Mana Dome only says that "No enemy spell or enchantment can target..." My understanding is that target refers to the initial casting. Otherwise the target of the use of an ability on an enchantment considered a target of the enchantment, which seems weird.