By a quirk of history, this particular culture won and imposed it customs on everyone else.
There's a societal down-side to polygamy, one that needs STRONG cultural overrides to prevent. If (presumably) richer men are allowed multiple wives, that means that there are fewer wives for the rest of the men. You then end up with an excess of unmarried, non-parental young adult men, and being married and a parent is usually a calming influence. These single men are usually the first in the streets if things take even a tiny down-turn. We still see this in Arabic countries which allow polygamy, as well as countries where there's an imbalance of men and women, such as China and India (one-child policies as well as gender-based abortions responsible.
This is an obvious problem in societies that also have the problem of being strongly patriarchal and/or misogynistic. (The obvious examples you site have these issues.) In cases where women are equally allowed and able to engage in such relationships, there is no a priori reason to suspect such a problem.
The only evidence I know of that is directly relevant to modern times and from a sexually equal setting is highly anecdotal. I've looked a little for better without luck. But, what I've seen and heard from the polyamory community is that this is most likely a non-issue, and that if it isn't, you probably have your genders reversed. Basically, I've seen weak anecdotal evidence that in some circles, the women tend to participate in more relationships than the men do. I haven't seen any evidence (weak or otherwise) of the reverse effect. And, of course, this report should be taken with a large grain of salt, as it's based on fairly strongly selection-biased sources. However, I think it's strong enough to call your fears into question.
For reference, I (male, straight) in a happily polyamorous relationship. My partner (female) has a paramour (also male, also straight). The three of us get along well, and none of us are actively dating anyone else.