Happy to be corrected, but I understand the first time this was mooted (Rags = Catholic, City = Prod) was Busby in a tv interview in either the 50s or 60s when he openly said "...are the Catholic club...".
My dad - schooled at St Francis, behind the monastery in Gorton - a Catholic and a Blue had never heard of this apparent religious link, as many Catholic, Gorton Blues of his age/era hadn't either.
Rangers have always had strong links with Chelsea and Everton. I think the so called link with City stems more from the 80s, or at least became more prevalent then, as we saw the multiple testimonial/friendlies/love-ins that lot were having with the Celtic. So rather than being a religious thing, it was more going opposite to that shower.
Often at games, in the Kippax, the shout would go up "Celtic!" followed by "Rangers". I couldn't tell you if it was 50/50, whether anyone was a genuine Celtic/Rangers fan or whether it was just shouted for the sake of something to shout. I think the latter personally as the shout was often followed by "Tiswas" with the response "Swap Shop!". Pretty certain it had nothing to do with defining certain sections of our fan base!!!
Whilst religion has certainly played a part in our early history being founded as St Marks, I am of the opinion (based on experience and what I've read) that as a club it has never really played the kind of significant role it has for the Old Firm and such.