Who or what is the 'God' of communism?

I suppose it depends which God you are talking about. There are some people who believe (or believed, when they were alive) that Vladamir Lenin is a sort-of God. In this case, yes, God would be a communist.

However, I highly doubt that many people would believe that the Judeo-Christian God (the God of the west) is a communist. Rather, this God is more of a king. Which, I suppose, would mean that He's a monarchist.