You're framing it in the American context though. Mixing politics and religion is taboo in most western democracies. And, religious conservatism isn't the same as political conservatism.

Ireland is a good example - devoutly Catholic but with very liberal politics. Gay marriage has broad support.

UK - pretty much godless and very liberal - but historically politics has been dominated by the Conservative party. Broad support for same-sex marriage, and it was actually brought in by a Conservative government.

China - authoritarian single-party system. Follows Communist ideology with a capitalist bent. Religion - taboo, government controlled, doesn't feature in politics at all. The important difference is that its culturally very Asian. Even eastern democracies like South Korea and Japan have a tricky relationship with gay rights.

If that were to change you'd imagine CCP would support gay marriage as it poses no threat to ideology or party policies.