Well, to start with, I wouldn’t actually say that humans have more relationship styles than any other mammal. I think that the issue is that how we look at human behavior and how we look at animal behavior are very different things- obviously, we have a much more detailed and nuanced view of our own species than any other.
It’s important to remember that most science is consumed by the idea of the average. That is, if we were trying to write a description of how an animal behaves, we’d describe what we saw happening the most often. This is a useful way to look at things, because it allows us a way to generalize populations. But this does not mean that that is the only way that things will happen.
If you tally up the human species by culture, you will find that about 85% allow occasional or frequent polygyny (the practice of one man marrying multiple women). That’s a huge percent, and if an alien researcher were examining human behavior and saw that statistic, they might conclude that humans are overwhelmingly polygynistic.
Of course, it’s a little more complex than that, because even among cultures that permit polygyny, most men only marry one woman. Generally, in these cultures, having multiple wives relates to a man’s wealth and resources. This is similar to mating behavior in, say, the red-winged blackbird. Males with the best territory will often attract more than one female. This also occurs in red foxes, which all the literature will tell you are a monogamous species: under the right conditions, male foxes may take more than one female mate.
We, as humans, are familiar with many more relationships within our own kind than monogamy and polygyny, of course, but that’s because we live and breathe human behavior every single day. We rarely spend so much time with animals, particularly animals that are breeding, which is why we might assume that their behavior is static or can be easily categorized. This is simply false. The “averages” that science looks for are generally the widest part of a vast spectrum of behaviors.
Normally, what we find in animals is that the degree of behavioral flexibility relates to how specialized an animal is. If a species lives in only one type of environment, with only a few resources it is designed to use, it may not need the behavioral flexibility that leads to multiple mating types. The opposite is true for humans; we are arguably one of the best generalist species around. We live in a myriad of habitats utilizing a myriad of resources and thus, arguably, have the capacity to accommodate a range of sexual lifestyles.
This is, of course, a gross oversimplification of human behavior, as everybody who is human knows. It doesn’t at all take into account the individual- their desires, their values, their history, et cetera. This is a degree closer than most science goes, and it goes for humans as well as animals. Individual animals will have their own behaviors and preferences just like humans do. We just look at them from much further away than we do ourselves.
Further Reading:
Different trends in human culture: Gray, J. P. (1998). Ethnographic atlas codebook. World Cultures, 10(1), 86-136.
Not only are red-winged blackbirds polygynous about half the time, they also show a high degree of promiscuity and polyandry: Westneat, D. F. (1993). Polygyny and extrapair fertilizations in eastern red-winged blackbirds (Agelaius phoeniceus). Behavioral Ecology, 4(1), 49-60.
One study found that about half of red fox litters were not the product of a monogamous male-female couple; both polygyny and polyandry occurred among the remainder: Baker, P. J., Funk, S. M., Bruford, M. W., & Harris, S. (2004). Polygynandry in a red fox population: implications for the evolution of group living in canids?. Behavioral Ecology, 15(5), 766-778.
Greylag geese most often form male-female bonds, but under some circumstances they may form male-male or male-male-female bonds: Sommer, V., & Vasey, P. L. (Eds.). (2006). Homosexual behaviour in animals: an evolutionary perspective. Cambridge University Press.
Marmosets and tamarins display a range of mating styles, including polyandry, monogamy, and polygyny: Goldizen, A. W. (1988). Tamarin and marmoset mating systems: unusual flexibility. Trends in ecology & evolution, 3(2), 36-40.
These are all just examples of animals with highly flexible mating styles that I came up with off the top of my head. There are many more species where differing mating styles are less common and therefore less frequently documented, but still exist.
Brinens and Things 
