You are correct. Judaism is rare in that it is at the same time a religion, a nationality, and a race all in one.
You can certainly choose to view it that way, but most Jews since the dawn of time have been Semitic, which is a race. Indeed the vast majority of Jews today, or anyone else for that matter, would say that Judaism is indeed a race. You say it is only a religion, but there is no question that it is at the very least also a nationality.