"Polyamory" is usually used to refer to having more than one romantic relationship at a time, which is pretty unobjectionable as far as the law is concerned. It might get you into trouble in a divorce proceeding, but the laws that still exist addressing it are largely dead letters.
I assume you're asking more about polygamy, or having more than one one spouse at a time. I was surprised to learn that there hasn't been much activity in this area since Obergefell, at least not a lot that has led to development of the arguments you're asking for.
The most on-point case I can find is Collier v. Fox, No. CV 15-83-BLG-SPW-TJC, 2018 WL 1247411 (D. Mont. Feb. 22, 2018), where a married couple and another woman sued the state because its criminal and civil laws prohibited the second woman from entering into the marriage. The court tossed the challenge to the criminal prohibition based on the parties' standing, saying that there was not a sufficient threat of criminal prosecution, and it dismissed the challenge to the civil restrictions based on Reynolds v. U.S., 98 U.S. 145 (1878), where the Supreme Court said that each state has the right "to determine whether polygamy or monogamy shall be the law of social life under its dominion."
There was no appeal, so the decision was never reviewed.
The only other case I've seen directly challenging the laws was Sevier v. Thompson, No. 2:16-CV-659-DN-EJF, 2018 WL 1378803 (D. Utah Jan. 26, 2018), where some anti-gay activists tried to overturn Obergefell by suing to force Utah to let one of them marry a computer and three of them marry each other. If you can't guess how this ends, I'll tell you that it does not end well. The plaintiffs all admitted that they didn't actually want the relief they were asking for, so the court threw the case out on standing.
The same dude also tried to intervene in a lawsuit (United States v. North Carolina, No. 1:16CV425, 2016 WL 7335627 (M.D.N.C. Dec. 16, 2016)]) over the North Carolina bathroom bill, saying that a man trying to marry a machine or marry multiple people belonged in the same category as a man trying to marry a man or a man who identifies as a woman. I'm sure you won't be surprised to learn that the court told him to go to hell again.
Because these were all tossed at the very early stages, there wasn't much development of the arguments for or against polygamy, though it was clear that everyone was basically talking about an extension of Obergefell's recognition that "the right to marry is a fundamental right inherent in the liberty of the person."
There have been several scholarly articles, though, addressing the issue. A few that popped up include:
- A Yale law professor considering whether plural-marriage bans could survive a challenge based on equal protection rather than due process;
- A Loyola Marymount professor reviewed several post-Obergefell books considering the future of plural marriage and concludes that (1) Western condemnation of plural marriage is strongly informed by racism and xenophobia, (2) legal reactions against plural marriage may do more to harm the people they are meant to protect, and (3) whatever benefits may accompany plural marriage, banning it does not entail the same kinds of harms as bans on gay marriage; and
- A William & Mary law student published an article running through the different ways a court might analyze the question and argued that Obergefell should generally permit polygamy, regardless of which test a court applies.