I went to law school in New York. Look at the state statute. Unless a state says specifically that a valid license is a prerequisite for marriage, it isn’t (in New York, a clergyman is subject to a fine, but that’s it).
The license is a revenue measure, but it doesn’t affect the validity of the wedding. The people who think they are “officially” single because they had a wedding without a license could be charged with tax evasion if they minimized taxes by filing separately (though in practice, most frum people gain a tax benefit from marriage).
Like much of the marriage law, getting married without a license is a feature largely ignore since it create more problems than benefits. Another ignored feature is the option of marriage by filing a copy of the marriage contract with the clerk’s office.