According to what God’s prophets say in Christian Bibles, I see Him encouraging people to reproduce, and at the same time assigning lifelong responsibilities to any man and woman, the first time that they join physically in the way that sometimes produces children. Those responsibilities include provisions that leave no doubt about who is the biological father of any child. I also see God prohibiting the practice of substituting men or animals in the place of women, for that kind of joining with a man, and the practice of substituting animals in the place of men, for that kind of joining with a woman.
Christian prophets always explain from the law about marriage. Everything is bounced off of that. Marriage is for more than reproduction. Its for happiness, peace, children and also for some other things.
In addition to reproduction there are some other reasons for marriage, some even more important than reproduction (debatable). A new husband stays in town for the first year (Deut 24:5), specifically to make the wife happy. It implies that there is a responsibility to make her happy at other times as well. Is that happiness more important than reproduction? Maybe. I am not in a position to debate about it, but I can see that there could be a debate. This may relate to your questions about same sex but am not sure.
So in addition to reproduction, marriage is about happiness. It can't be just any marriage but should be a happy one. The prophets expand on this.
Marriage is also about non-violence. In the biblical times marriages are arranged, and this is an important matter for bringing people together. In addition it combats violence another way. The laws mention a spirit of jealousy that comes upon men, seemingly for no reason. They get jealous, and their jealousy is violent in nature. Several stories have this violent jealousy in them. Marriage is expected to play a part in damping that violent nature. Its designed to be a peacemaking influence.
Jesus lays a lot of responsibility upon the male to make the marriage work. Someone (above) mentioned Jesus comments about divorce and eunuchs. What's happening there I think is Jesus is saying that even if the wife won't have sex with her husband, maybe he should still not divorce her. The term 'Eunuch' there probably means that if he's married and she won't have sex with him, he's a eunuch unless he divorces. The law specifies the man must be available at least monthly for his wife for sex, but it doesn't specify that she must be available for him (though that might be a provision in marriage contracts). Jesus opinion here is the man should win her back in this case. Those he's speaking to do not like what he's saying and think if he's correct then its better not to marry. Its not clear how they believe marriages ought to operate, but it appears that they do not put as much responsibility upon the husband to keep the marriage happy.
I don’t see any prohibition against two men or two women calling their relationship “marriage,” but I don’t see God assigning the same responsibilities to them that he does to a man and woman when they join physically in the way that sometimes produces children.
Most of the laws are for Jewish men, and its men who are supposed to know them and teach them to the children. This affects Christianity. Yes, technically the sex of the parent has something to do with who does what in Jewish society. There's a typical mom/dad scenario.
Marriage is also about bringing men to their children to educate those children, father to child and not mother to child. This affects Christianity. Whereas a man tends to want to go off to be alone, let the wife take care of the kids and only sometimes see the children, the laws specify he must teach his children. I think this is indirectly why we read Paul (Christian) saying "I do not allow a woman to teach a man" to Timothy. Its a difficult passage. Paul also says that in Christ there is neither male nor female, so how else to account for this statement about women teachers when there is neither male nor female? We're no longer talking about Jewish scripture, yet the teacher adopts the student into their family. We see, as with the Jewish people, Paul wants the Christians to have men doing the teaching. The next question then is why is he sticking to this family arrangement with the teacher being like a father? I don't know all of the answers.
I imagine a teacher-student relationship is intended to be a much more intimate arrangement than having a lecturer or just being an acquaintance of the teacher. I am not sure exactly how this affects your thoughts on same sex marriage, but in the Christian writers there are recognized differences between the sexes when it comes to teaching. It pays to be aware of it when you are considering questions about same sex relationships and what Christian prophets might have to say about them. They don't say a lot on the subject, but there are clues about what might impact the subject.