Nice observations. As an American, by the way, I am endlessly amused by my same-tongued siblings abroad who conflate school with a pursuit for small boys and girls in pigtails.
Here, it’s an umbrella term for any sort of education. Even a PhD candidate may casually say he’s on his way to “school” if he has a lecture to attend or even to give.
But given Matt’s photo, I imagine you were inclined to forgive him his language peculiarities. ;-)