Uh, I think he meant “job”. That said, I have never ever had to provide my Selective Service number, when applying for a job, and I have had jobs with pretty heavy clearances.
I've provided my DD-214, not Selective Service card since it would be irrelevant, but I do remember draft status being verified for those who didn't serve. I suppose that died with the draft, and it's a bit irrelevant. Someone younger might help here, but I believe they're required for college admissions, or perhaps if receiving college loans now. And in those days employers did health checks too, whether of not job related.
Still, were I a politician, it's the kind of thing I'd release if asked. Same with health records, and his Bar application, which would list all non-traffic infractions.