Succession war risk has to be handled somehow anyway because of multiple heirs and serial monogamy.I'm aware that most of the time polygamy is damaging to society especially if it's the king or Emperor who is doing it(because that is a recipe for succession wars) but even for commoners it's bad because humans have a gender rate of 50/50 between men and women. So if one guy has more than one someone else would have none and men not getting married and having no women is bad for social stability.
It doesn't mean that a man wouldn't want a variety of women(and while it's wrong to fornicate there are situations where polygamy can be allowed and it won't damage society and will in fact help. Like if there was a massive war that cut down the male population then you'd want women to not be alone and have a man to support them and the population to grow) this is rare though and it shouldn't be common because near apocolyptic wars are a bad thing. But generally polygamy is not an inherently wrong thing like sodomy for instance. Because there is never an instance where sodomy is helpful, but there are small circumstances where if you have a low population of males it would be a good idea to have polygamy for only one generation.
It's also not exactly common men who have multiple wives in the Arab world and Muslim Asia, it usually is only the rich people who can afford that, though in some shitholes in Africa it can get big enough to have major demographic effects.
Another effect worth mentioning is that combined with the region's tendency to cousin marriage it can create some seriously confusing family trees, especially back before the days of good record keeping and basic genetics.
And also as mentioned, it creates an international pressure, not just into wars, but also for importing foreign brides if there are poor countries accepting such polygamy available.
Wonder if there was ever a culture that got really mean to the neighbors with it and set up a hybrid system that allows only monogamy internally, but foreign wives don't count for the limit. Rome may have something in the direction that idea with slavegirl relations for men not counting as adultery.