Your three components of marriage highlight how much marriage is a social institution and not a "right".
Rights are for individuals, God given and exist independently of society -- such as the right to free speech and the right to peacefully assemble. These do not require anything from society or other individuals except non-interference.
Marriage, on the other hand, is the establishment of a public social partnership, in a public ceremony, requiring public sanction, conferring public benefits, and requiring others to grant the married couple special privileges and treatment.
About as far from a right as you can get.