Mine does, issued by Illinois in the 1950's. My wife's does as well, from Ohio during the same decade. My daughter's doesn't, but her's was issued by Illinois in 1991. It wasn't at all uncommon for birth certificates to list "Race or Color", as that part of the document puts it, yeaws ago. That has changed over the last 20 or 30 years.
This document isn't a birth certificate. It's a certification of birth, I'm looking at my certification of birth right now, and it doesnt show race...... but my birth certificate does.