Specifically, what would be its significance, in the law? In what respect would it make a difference, in the law?
Certain unions in addition to those traditionally recognized between a man and woman would be recognized as having certain legal characteristics. Certain legal rights and obligations between the partners, and the between the union and third parties, would apply in the case of the union, or partnership (which is what a marriage is, as to money, property and contracts) of two humans of the same sex in addition to two humans of the opposite sex (why "opposite", by the way?). There would be more work for lawyers (we keep creating such work, on an almost daily basis, and yet complain that there are too many). More work for the courts, as there is no reason to believe such unions between two of the same sex will be any more successful than those between two of the opposite sex. And that would, as they say, be that--for the law.
There are dangers in legislating morality. Sometimes, morality figures in the creation of law. The law, once in place, is not moral, however. It is a functioning system, often of great complexity.
The law shouldn't concern itself with the morality of marriage. Let religions treat as "marriage" whatever they deem appropriate. Let the law treat "marriage" as it already treats it--as a partnership, nothing more, nothing less. If ideas associated with the word "marriage" create problems in the law, eliminate it from the law, and let all such unions be called "domestic unions" (or partnerships, or whatever) and be done with it.