There is simply no way to legally recognize poly* relationships under the law in a way that resembles binary marriage.
Marriage is a deeply troublesome arrangement, with a storied history of abuse.
Yes, there is probably more abuse in marriages then outside, but this is not an argument against marriage, but against abuse.
And where have we heard precisely these arguments before?
People are great.
Polygamy is 1 man to multiple women. Polyamory is between more then two people of any gender or combination of genders.
But you're also wrong! Polygamy refers to any marriage that has more than two partners, gender identity is irrelevant.
One man many women is polygynyz one woman many men is polyandry. Polyamory is multiple partners.
I think the more likely path than legal polygamy, though, is to expand a more flexible set of legal arrangements for families, as a replacement for traditional marriage being the legal framework. The fact that a huge proportion of couples here (Denmark) no longer get married is already forcing that for another reason. Since many families which otherwise look traditional — long-term cohabitation of an opposite-sex couple, jointly raising children of which they're parents, etc. — don't get married, marriage as the organizing principle of families is becoming less relevant, so the law already has to start handling things differently.
"De facto unions" have existed (for both hetero and homosexual partners) for more than ten years already, and have many of the same rights as married couples, including inheritance, refusal do testify in court, etc.
As is traditional marriage, but to a much larger extent.