There is a consititutional separation between the government and organized religion.

This is simply not true. Separation of church and state is not in the Constitution, which only says that the government shall not establish an official religion.

The actual text of the relevant portion of the first amendment is: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof". There are various interpretations of this. For example, if we read the Letter to the Danbury Baptists by Thomas Jefferson, you see that he intended separation of church and state.

Theoretically, the free exercise clause seems like it would be enough to change the situation. After all, doesn't being legally forced to follow the rules of one religion violate one's rights to freely exercise their beliefs? Though there's not a lot of reason to expect it, after all, the government has eliminated the rights of Mormons to practice bigamy, which would be (or at least would have been, when it was officially condoned by them) a violation of their religious freedom.

And besides, isn't the idea of marriage religious in nature? Aren't they preferring one form of religious belief over another by sanctioning a practice that is not common to all religions?

This means that the government cannot choose to support one religion over another. However, the Constitution does not prevent the government from giving benefits to religious groups, as long as it gives those benefits equally to all religious groups.

Are you suggesting they should be able to give preference to belief over non-belief? Fortunately, the supreme court has, at least part of the time, decided that non-believers should be entitled to their (lack of) belief, and should be treated equally.