The arguments against multiple marriage (i.e., polyamorous families) are similar to the arguments against same-sex marriage that the US has rejected.

One traditional reason, in patriarchal society, why most men opposed polygamy (which always took the form of multiple wives for one man) was that it implied some men could not have wives. In effect, banning polygamy was a measure to improve equality among men.

But nowadays, as rich men have many ways to have multiple lovers and women have equality at least in principle, I don't think this ban achieves its goal very much any more.