Marriage is not a religious institution. It predates the Old and New Testament. Genesis 2, when the first couple was married, happened before mankind even needed a Savior.
If marriage were a religious institution, then non-Christians, Muslims, Buddhists, atheists wouldn't bother to marry. But they all do. Because the instinct to marry is a part of our nature. Marriage is a "natural" institution, not religious like Communion.
Government recognizes marriage because of very real legal issues that arise when two people come together. There really is no other alternative.
So government, by continuing to use the definition of marriage that it always has, is not "mixing religion and government" unless every government on earth does throughout human history does. All we're asking is that government continue to do what it has always done - nothing.
It can be both. Natural marriage serves to promote the earthly good of husband, wife and children, and therefore of society. Sacramental marriage is ordered toward the eternal salvation of all of the members of the family.