That is my understanding of older certerficates of live birth, especially where the info was derived from births in religion based hospitals. I have no direct evidence of that as I have never had the opportunity or reason to review any.
Don’t know about religion-based hospitals, but I’ve seen several American birth certificates back 100 years or more, having done some family history research, and none of them had fields for religion. I know that Hawaii had no such fields on their long form in 1961. I’d be curious to hear if anyone has seen a birth certificate from any state that had religion data on it.