"They imagine all sorts of nonsense regarding giving up rights, but the reality is that the government is serving as a repository of the covenant."
License is a prior restraint. It presumes that the activity in not a right but stems from the licensor (government in this case), is illegal otherwise and can only be done with permission of the licensor. Is this the case with marriage?
The government does have a right to decide who it recognizes as married since it gives special privileges to those who are married. "The American colonies officially required marriages to be registered, but until the mid-19th century, state supreme courts routinely ruled that public cohabitation was sufficient evidence of a valid marriage." [Stephanie Coontz, "Taking Marriage Private," New York Times, 11/26/2007]. Government ought to restrict the right of marriage to only those who have the right to marry in God's sight, but government being what it is, often includes and excludes more than it should.
For example, there was a period of time in United States history when people of different national heritage could not marry. In such cases, the government is violating the laws God gave them to uphold (Romans 13:1-2) and Christians follow God's laws before man's laws. We now have states which grant license to married to homosexuals, but this is a violation of God's laws as well. Christians will not recognize such unions since the people are not married in God's sight.
It is called a marriage license because the state attempts to make some determination as to whether a couple has the right to marry. The government has age restrictions, which do not interfere with Christian principles. It restricts marriage to a single partner at a time, which is something Christians find agreeable. The main grievance Christians have with the government at the moment is that the government is too accepting. From a Christian's point of view, the government merely serves as a repository for the covenant made. Getting a marriage license does not violate any law of God and since we are commanded to be obedient to the government when possible, we should abide by those laws.
What I was referring to in regards to giving up rights are people who think that signing a marriage license gives the government control over your children [See "Can I tell you about a way to get married without government involvement?"]. These are baseless fears.
Yes, a license does restrict some rights. Two fourteen year olds can't get married. A person can't have multiple wives or multiple husbands. None of these restrictions on a recognized marriage are disagreeable to Christians.