The Tennessee Waltz was my grandmother's very favorite song! What an interesting question. You are right, traditionally a man is introduced to a woman; however, in this case I am quite sure that the author sacrificed any idea of remembering etiquette to the rhythm of the lyric, and nothing subtle was intended.
I don't know about GB, but in the US I don't know many people who scrupulously follow this rule---maybe it's just that I don't move in the right circles. :) And in business situations, of course, the only thing that matters is who outranks who. Gender is ignored.