I wasn't familiar with Johnstone's topological topos but yes, it's exactly that.

@R.vanDobbendeBruyn thanks, I was worried about that (I copied this from a document that I wrote so it's longer than an MO answer). I split the axiomatics into another answer and added some more intuition to this one.

@ZhenLin I've glanced at the definition but couldn't parse it right away

To me, this is an argument in favor of rejecting AoC, since it isn't necessary for the things we actually care about. I could just as well say "nobody cares about the crappy set you slapped together which isn't measurable".