I think we should approach this the other way: take the word 'marriage' out of the law, and replace it with something like 'civil union' - then you can call your own relationship whatever you want. Other than looking at how the benefits and assets get divided, it's not the state's business.

As for the religious angle - I'm not sure how the Church got control of the marriage business, but I suspect it was something like this: some cardinal realized that if the church were able to get a few cents, pence, or drachma for every new marriage in Europe, he would within the year be able to finance the construction of a villa for his latest whore - so the word went out: be married by clergy or be burned at the stake.