From the article it seems like the US States got into the marriage business for reasons of racism/eugenics:
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. By the later part of that century, however, the United States began to nullify common-law marriages and exert more control over who was allowed to marry.
By the 1920s, 38 states prohibited whites from marrying blacks, “mulattos,” Japanese, Chinese, Indians, “Mongolians,” “Malays” or Filipinos. Twelve states would not issue a marriage license if one partner was a drunk, an addict or a “mental defect.” Eighteen states set barriers to remarriage after divorce.
and now it's all about control; you can't get your "benefits" if you don't have the government papers.
As Nancy Polikoff, an American University law professor, argues, the marriage license no longer draws reasonable dividing lines regarding which adult obligations and rights merit state protection. A woman married to a man for just nine months gets Social Security survivor’s benefits when he dies. But a woman living for 19 years with a man to whom she isn’t married is left without government support, even if her presence helped him hold down a full-time job and pay Social Security taxes. A newly married wife or husband can take leave from work to care for a spouse, or sue for a partner’s wrongful death. But unmarried couples typically cannot, no matter how long they have pooled their resources and how faithfully they have kept their commitments.