The key question is the following, to wit:
Does the State not have an interest in knowing who is married to whom for the purpose of knowing who is the parent of whom in order to keep track of who owns what, and what passes to whom upon death and inheritance, and so-forth?
In order to do that, the State has to know who is having children together. Part of the reason polyandry was in most societies considered abomination is because no one knew who the father was. Of course, with the advent of paternity tests, that is not a problem in advanced countries, although it still would be in many.
I think that is the whole reason the State should be involved in regulating marriage, is in order to keep track of who is with whom, and who is inheriting what, and who is giving birth to whom, and so-forth. As to who is marrying whom, that is a totally different story. As long as the Churches, Synagogues, Temples, Mosques, et al of the land are not required to endorse things that go against their own beliefs, if you want to marry your lawn chair, and said chair is sentient and adult and can consent, feel free.
No, in all seriousness, with the exception perhaps of polyandry, I think humans ought to be free to marry whom they wish, at least at the State level. I might even be able to accept polyandry in those societies advanced enough to maintain records on paternity.
Obviously, my own religious views are different. But I don't believe in imposing my religious views on others.