Hi John,
"Sobriety" did have the religious aspect that you described in your OP. "Sober" and "sobriety" are used quite a bit in the King James bible. It refers to behaviour and dress, etc. I would imagine that within certain groups in UK society it might be possible to hear people still using the word in this way (but as you said it's rare).
It seems to me that the idea is very closely linked to the beliefs of many of the early religious settlers who left Britain seeking religious freedoms in America. Given that the USA is a more religious country than the UK, and that the population is much bigger, I wouldn't be at all surprised if a large number (but maybe not a large percentage) of people do use the word with the religious connection. In fact, if you google it you'll find some nice internet rabbit holes to go down. I think it's an example of a sort of sociolect meaning.