Well, I wouldn't doubt that the army teaches that because "soldier" is the specific term for a member of the army. It's self-serving for the army to do that.
But I've never known a Marine, sailor, or airman who considered themselves a "soldier".
I agree that any marine, sailor, or airman would shake their heads, and they probably should, but I remember it clear as a bell. I think it was even a question on an E-5 promotion board.