I wasn't necessarily referring to evolution, but evolution is probably why my brain chose monkeys as the example. I understand what you mean, I just don't know of any examples. There is value in observing animal behavior so that we can deal with them properly, I've just never understood how it could shed light on our own behavior.
Some things we learn from animals include which foods are probably safe for ourselves, and where danger lies. Dogs make excellent hunting partners, sensing and doing things that we can't. The very first canine movie star was a huge celebrity, often charming hundreds of visiting fans per day. His movie company took him to a board meeting to show him off, and he instantly lunged for one board member, who was later discovered to be embezzling. My friend Don and his buddies were camped on a remote coast, and low on food. Their dog Walter had seen how the Wolves would chase deer into the water, and soon chased one himself so his friends with canoes could finish it off.
When we see squirrels hiding nuts for the winter, it reminds us to do the same, and when we see them re-hiding them, it reminds us to watch out for theiving spies. When the birds go silent, we know to watch for a predator. If we learn to co-exist with bears, we also learn how to get along with moody, strong people. When we see our pets getting jealous of each other, it reminds us that we often do the same in less obvious ways.
A.V. Roe used to take his vacations where he could get close to cliff-soaring birds, and then helped to invent the airplane. My tree surgeon once supplied a Beaver lodge with fresh food one dry and dangerous for them winter. In the spring, a Beaver walked right up to him and his lady, and dropped a green branch at their feet to say thanks.
Sailors used to follow birds to find small, isolated islands. A Dolphin named Pelorous Jack used to pilot ships through treacherous Pelorous Strait. Watching the mating behaviour of animals, and how they cooperate as parents can often help to illuminate and explain our own, less obvious or frequent behaviours. Observers of Carmine Bee-Eaters, who nest near each other on cliffs, say that it is like watching a popular TV series with frequent infidelity, sped up a thousand times.