In any culture where polygamy is a long standing tradition, it's still quite rare. Only the top wealth level men have more than one wife. The rest of the population have conventional one wife households. Kinship by marriage is probably the oldest form of inter-family alliances. It's not difficult to see that the man who controls more resources will have other family groups who want to form alliances, and limiting yourself to one wife would block a lot of lucrative opportunities. On the other hand, the guy with four sheep has to make do with whatever family will let him have a daughter. He knows better than to ask for a second. It's a pretty stable system. Everybody knows their place and raising one's status by marrying up is one of the few options.
When modern societies adopt polygamy, it's usually because a shortage of young men and an over supply of women of all ages. Polygamy seems a reasonable solution, even though implicit in this is the idea that women cannot control resources for themselves. They have to be under the roof of a man. Also implicit is the idea that a woman has little say over whose wife she becomes. Seriously, why would an 18 year old(or younger) woman want to go be the third wife of a 60 year old man, and his first and second wife. It's a short term solution which quickly outlives its purpose, when balance returns to the marriage age young men.
Once again, marriage becomes a way to form economic family alliances with powerful men. What is observed is, wealthy men with marriage age daughters trade them with similar men. The young men of the culture are caught in a bind. If they want to marry, they have curry favor with the guys who have control of the young women. The rebellious teen age boy not only cannot get a wife, he can't even get a date. By the third generation, it's become a modern feudal society, where everyone owes their livelihood to someone who has more wealth and power, except in this case, they're all old men.
The result is a lot of young men desert the culture. This is inevitable when more liberal but monogamous societies are withing walking distance.
Marriage, as it exists today is a special form of legally sanctioned corporation, where the principle stockholders are expected to have sex with each other. The law defines the aspects of the corporation and the responsibilities of the two stockholders. If a child of the marriage is not properly clothed and fed, the government looks first at the names on the marriage license. There are many more such things. Now that women have all the legal rights as any man, it's reasonable to ask if monogamous legal marriage has any relevance at all.
There's nothing to prevent a man from legally marrying one woman and then bringing home as many common law wives as he can manage. In a society where women have all the legal rights as a man, this may be as difficult as it sounds.
As for the question about multiple girlfriends, how does that work? I've been in relationships which were exclusive to a certain degree, for the past 50 years. One thing I learned at an early age, I could have two girlfriends, only as long as neither of them knew the other existed. I think multiple girlfriends violates some law of physics, or maybe one of those rare man made elements which are created in an atom smasher and exists only for a few milliseconds.