What is the classifier môn used for? The sentences I saw it paired with include one about playing a sport and one about philosophy. I suspect something like "subject, lesson, sport (school)."
Môn = course, path, track.
chuyên môn = specialized in, something you're good at.
specialize in pickpocketing = chuyên môn về móc túi.
You are correct in that we use it mostly for academics and sports. You can use it on the individual course or a track.
For a career field, we use nghành. In English, you would never say the "tennis field." So it doesn't matter whether you play tennis professionally, you would never say nghành tennis, but you say nghành thể dục thể thao (sport field).
So Nghành is broader than môn, the same as field is broader than track.