The allowance of more than one marriage partner in today's equality of sex environment ( as opposed to the plurality of wives only in the past ) would then allow difficult situations. Man A marries woman B, fair enough. Man A then marries woman C which again is fair enough. But then woman B marries man D. So is man D then married to woman C, no, not really. But then man D could marry woman C. Things start to become very convoluted and the more partners that are introduced the more complex the situation becomes. Introduce adult children after long periods of marriage and suddenly if a daughter of the A-B marriage wants to marry the son of A-C marriage and now does this become incestuous because of the common father? And what about children with a common mother - would that be incestuous? Ok, maybe hypothetical but if it's possible to happen it probably will. So for this reason I don't think it should be allowed to happen.
In my opinion, limiting an individual to only one marriage license is a direct conflict with the Supreme Court ruling, Lawrence v. Texas. The private matters of consenting adults behind closed doors shouldn't be intervened by the state. Limiting an individual to only one marriage license is the same as saying certain sex acts aren't permitted with your spouse.
Marriage should be allowed to only one person, and not any more then that. You can not truly love two people at the same time, and the one who is loved less should not have to deal with that. Allowing marriage to more then one person is a slap in the face to people in real marriages.