Sure. Perhaps you've heard of bigamy? Alice can't marry Carol because Bob already has a vested marital interest with Alice. For example, if Alice marries Carol and dies, Carol is entitled to 100% of her assets as spouse. But so is Bob.
That's not the policy rationale for the prohibition on bigamy, and while it is perhaps a little better of a reason than administrative convenience, it boils down to the same thing, since the question of marital property is one of the issues that legislatures will have to address when the ban is overturned as it inevitably will be.
On the contrary, tradition is absolutely relevant as to whether something is a fundamental right. Marriage is a fundamental right because it's enshrined in our traditions and collective conscience. ...
Polygamy does not have such a place in our traditions or collective conscience, and therefore is not a fundamental right.
Yep, that's the bullshit argument that people were rolling out against same sex marriage all right. That because it wasn't traditional, it wasn't fundamental.
The core mistake with that argument, whether in the context of same sex marriage or marriage among persons already married, or in larger numbers than two, is that what's fundamental is not opposite sex marriage, or same sex marriage, or polygamous marriage, but simply marriage, without qualification of any kind.
Issues like gender, race, consanguinity, marital status, and number of spouses are all restrictions on that singular fundamental right. Whether they stand hinges on whether they can be justified. Two of them, it transpires, cannot be. Ultimately I think the only restriction that will hold up will be consent, and perhaps consanguinity will have to be reframed in terms of consent if it's to be salvaged.