Polygamy has been extremely common and prevalent throughout history across time and cultures. However, with only a tiny handful of exceptions, it has always been one man with several women. Even if having multiple legal spouses wasn't supported by the culture or society, rich and powerful men regularly had multiple mistresses, concubines, prostitutes, slaves, etc. Some recent genetic studies suggest that only 10% of men throughout history have passed along their genes, while almost all women who survived to childbearing age did, implying in most societies a handful of powerful men had access to most of the women.
Biology was the basis for these ancient arrangements--a human male has to invest very little to reproduce, and can fertilize many females over a short period. A female can only be fertilized by one man at a time and invests the greater part of a year carrying offspring to term. Biology doesn't restrict us in the same way these days, but that's the early foundation of human relationships.
In the Western world, most of our nations inherited the foundations of civil law from ancient Rome, and the Romans had very clear laws about monogamy (and divorce). Why?
The purpose of marriage as a legal institution is because even rich people can't make use of their property after death, so they wanted to ensure it would be passed on to appropriate heirs. For the Romans, children born within a marriage were "legitimate", and thus entitled to their parents' property. Children born out of marriage were "illegitimate" and entitled to nothing. Thus, a Roman patrician could have affairs with as many slaves or concubines as he wanted, but those children wouldn't inherit a thing--only the children had by his lawful wife would matter. So instead of having a dozen children fighting over their parents' estate, the Roman legal authorities could just declare the legitimate children as the heirs and tell the illegitimate ones to get lost. (The legal heirs might still fight among themselves but at least you've cut out a lot of the conflict.)
Religion might have invested the institution with all sorts of moral reasoning and whatnot, because in earlier times the boundaries between law, religion, philosophy and science weren't as strong as they are today (Roman "prayer" was more like a legal contract between the worshipper and the gods). But religion strictly had nothing to do with the original intent of marriage: a legal measure to ensure that property passed smoothly from parent to legitimate child without strife.
And that's why we don't do polygamy in the Western world. Even though we've invested a lot more meaning into marriage, like love, religion, family, etc., at its heart it's still a legal arrangement for the sharing and distribution of property, and adding more people to the contract would make things infinitely more complicated for tax purposes, family law, power of attorney, etc.
