I'm guessing the "AI" was used in the geometry mapping from the sat data they used for their region maps.

Artificial Intelligence could be used for a lot of things, like:

weather generation

ground handling at airports

random bird strikes

controlling in-game aircraft and their routes

Realistic ATC that understands what you say (think Cortana) and responds intelligently.

In the trailer, I saw multiple planes flying alongside. I was wondering whether this meant multiplayer (no idea if this is common in flight sims, they're a genre I still want to get started in), but it could also be a computer that flies alongside?

Azure AI is Microsoft's artificial intelligence platform in the the cloud , running on Microsoft's Azure cloud platform. I am guessing they will be using a cloud based AI service to provide AI features in the game.

Nothing