Why government should be involved with marriage: First, people are going to engage in sexual relations with each other. For the vast majority of people, such sexual relations can and often do result in the conception of a child. Allow me to exclude from this assumption engagement in “alternative sexual acts” however common they may be, exclusively gay persons, infertile persons, women past menopause, etc. I’m making a gross generalization with an awareness that there are many (though overall still a minority) exceptions to it. Among a large percentage of such persons, one or more children are wanted.
Society considers itself responsible for assuring that children too young to work receive proper care to enable them to grow into the next generation of productive adults. (Note that this does not presuppose government responsibility to provide or finance that care. Government may be libertarian, depending on charitable means to provide it, or may mandate that parents provide it.)
It is socially beneficial that every child have a caring nurturing parent (whether biological or adoptive) providing for it. Given our social structure, there is a strong advantage to having two adults providing that care. This corresponds well to the two adults it requires to procreate.
There may be circumstances (“deadbeat dads,” “unfit mothers,” etc. – see any daytime talk show for examples) where the state is obliged to step in and arrange for a child to receive that proper care. But in general, it is considered far more advantageous to give legal recognition, and often tax advantages, to the desire of a couple which wishes to spend the remainder of their lives together with appropriate boinking and consequent offspring. This assures that there are two adults present and providing for the care of the child(ren) which they have produced, at little or no cost to the state.
Marriage is not exclusively religious: There is absolutely no correlation between the desire of two people to commit to each other and their putative religious faith. Atheists fall in love just as often as fundamentalists, and every gradation betwen. However, marriage is seen by some faiths as entailing a religious commitment. One makes promises before God, the gods, the tutelary god of marital unions, one’s ancestors, or Whomever, relative to the union one is committing to in marriage. These commitments are similar to the ones the state tends to seek the marrying couple to make to assure the care of the resulting offspring. So there is no clear and practical reason why the religious officiant (priest, pastor, rabbi, bishop, whatever) should not function as agent of the state in receiving those commitments. This meets the Lemon test, in that there is a clear secular purpose distinct from the religious one, and far from entangling government in religion, it reduces the duty on government by enabling the clergy to assume a share of the duties of the marriage-officiant civil official, as regards their own co-religionists.
Marriage requires some degree of regulation: Children become capable of receiving sexual pleasure at a much younger age than is required for mature commitment to a lifelong marriage. For most people this is the early teens; for a substatial minority of people, sexual arousal, sometimes even with orgasm, occurs before full puberty. Childhood/adolescent crushes on adults are a staple of human experience. Adolescents pair-bond and proceed as far in sexual relationships as they consider socially acceptable (note, not as far as their parents do, as far as they do). But these relationships are not (usually) stable lifelong commitments, but temporary emotion-based reactions. The result is a very fertile breeding ground for pedophilia and for unstable underage marriages. So obliging commitment for marriage to require a certain degree of maturity (generally an age of consent, often with parental or judicial signoff required for marriages at a younger second tier of ages) is beneficial to the goal of providing stable adult parentage for minor children.
Marriage does not require fertility. The array of factors that lead a couple to desire to contract marriage is as manifold and rich as are the varieties of human experience. The ability to jointly beget and bear a child is not a mandative factor. It is here that we bring back into the discussion those people we excluded from the generalization in the first paragraph. Because adoption, fostering, and other modes of provision for children by others than their biological mother and father, are fully acceptable ways of handling society’s requirement that children be provided for. And some couples will wish to take advantage of the nature of the marital commitment towards each other without any desire to add children to the mix.
Hence: The nature of pair-bonding, and the social requirement to assure children are provided for, make marriage an institution which requires to be encouraged. A delimitation on the basis of maturity is a good one. But other limitations are mandated by personal (often religious), not social definitions of marriage, and need not be applied. (I call on all participants to place tripartite and higher-level marriages and bigamous concurrent marriages in a separate thread, to avoid confusing the discussion with the sorts of extraneous complications such relationships necessarily entail legally.)