I have that skill to a limited nature. If I turn on the classical music radio station down here, and if the piece is by Vivaldi, Bach, or Mozart, I usually know it's by them almost immediately, even if it's not a famous piece.
I've had a crash course in musical listening since we joined this parish! Our prep school attached to the parish offers college level courses in conjunction with Spring Hill College in Mobile - and our choirmaster taught one, "History of Western Church Music." It was astounding - so far as I can tell, he knows everything about everybody from Mr. Ockeghem forward, and we just lapped it up. The man is a genius. I can't remember half of what he said, but even that puts you way ahead of the game.