A common law marriage just means a couple who live together monogamously - it doesn't convey any special rights.
No. In a different jurisdictions it confers various rights and obligations to the couple, from absolutely nothing to the vast majority of those of a formally married couple (of course, most jurisdictions don't recognize it at all). And you will notice that there is not one reference to monogomy on that article.
I can be monogamous with someone without living with them (therefore not be marriad by any common-law standard) and I can live with someone while having a sexual relationship with them without being monogamous (and be married by common-law).