The official AoE3 website claims you can play "multi-player campaigns":
Play against your friends
There are several multi-player campaigns where you can get into some heated battles with your friends.
However, the only campaign I've found in the game was the single-player one. The very "Campaign" menu option is only available after you click "Single Player" first.
There are multiplayer scenarios, but I don't think they mean that. The game itself distinguishes between "scenario" and "campaign", so does the official website, I guess.
So what are "multi-player campaigns" and how do I play them? I use "Age of Empires III: Complete Collection" from Steam.