How do I remove all traces of my Apple ID from a borrowed MBA?
Apps downloaded from the App Store are tied to the Apple ID they were purchased under, and can only be updated with that Apple ID. From what you're describing, you must've installed some apps (licensed to your Apple ID) on his Mac while you were using it.

The simplest solution is to figure out which apps are licensed to you, delete them from his Mac (you can simply use the Finder to drag them from the /Applications folder to the trash), and then (if appropriate) purchase/download them under his Apple ID.