I don't know when the phrase "em dash" got popular. It was probably due to web development, because, unless you were into typesetting, nobody knew what "em" was. We always just called them dashes--two hyphens make a dash.
I would go back to the advent of Desktop Publishing. The early Macintosh + Laserwriter really did a number in bringing esoteric terms like "font" to us commoners.
Some of us found out we were typography nerds and didn't know it until then.
