It is one factor.
Money as such is not that old on the scale of human evolution. Until quite recently value was exchanged directly. Even with coins they were exchanged on the basis of the value of the metal they were made of. And coins were invented only about 5000 years ago.
What you are referring to is commerce. That is, exchange of value for value on a voluntary basis, with the expectation (possibly enforced by a watchful eye and a hand on a ready weapon) that the other person will not cheat. Trading is evidenced much farther in the past than coins. For example, beads made of coral and other ocean products are found in grave sites many 100s of km inland. Such items are noted in sites dated to the early stone age. So, in the ice age more than 10K years ago and possibly 100K years ago, humans were already making trade goods and transporting them long distances. (There is also an interesting discussion to be had on the things we learn about humans from finding valuable trade goods in graves. But that is another topic.)
I have been unable to find any group of humans that do not trade, at least to some degree. It is a question of different levels of availability of material and skills. The folks by the ocean have lots of ocean products and not much of what is found in the mountains. The mountain folk have lots of mountain products. They each learn the skills required for their own products. They are much better off with a trade. They are better off with trade than they would be trying to steal the other group's stuff.
This is a thing that is not strongly shown in non-humans. It is not unknown but it is much less strongly displayed. This has overlapping reasons.
Animals have little ability to transport potential trade goods. A wolf pack that is near a large group of one type of prey animal might well be better off if they could trade with another pack near a different type of prey animal. Birds for caribou for example. But it is difficult to transport any significant amount of food when your only means of carrying it is with your own mouth. You can just about carry enough to feed a couple babies.
Also, most animals have little ability to store any large amount of food. Bees can store honey. Hibernating animals can live on food energy stored in their bodies. But these are not easily used as trade goods. So, even if trade goods could be transported, it is difficult for animals to build up a stock of trade goods.
And since it is very difficult for them, they don't tend to show any tendency to do it or develop any skill at it.
The beginings of trade are present in animals. There are courting gifts, gifts expressing solidarity to the troup, dominance/submission gifts, and so on. And there is reciprocity in such activity. It is difficult to call this commerce, but it is clearly on the path.
In the book Systems of Survival by Jane Jacobs, she explains the thesis that humans have two patterns of getting a living. She calls them syndromes.
One pattern she calls guardian. This pattern is based on getting hold of and keeping control over territory. The guardians then obtain their living by extracting it from the territory. Guardian ethical rules form a self-reinforcing system. It includes such things as honor, loyalty to the group, taking vengeance, suspicion of strangers, tradition, fatalism, prowess, and shunning trade. This is typical of the ethical system that hunter-gatherer nomadic tribes follow.
The other pattern she calls trader. This pattern is based on producing valuable items and trading them with others. This is intended to take advantage of the type of differences involved in skill and resource availability. Traders have a different self-reinforcing ethical system. It includes such things as gathering wealth (of whatever sort is possible, such as improving farm land), being optimistic, valuing liesure, being open to new things and new people, and shunning force. This is typical of the ethical system followed by villagers where trade is a large part of them getting their living.
Jacobs gives as one example, the transition of a tribe in Africa from hunter-gatherer nomads to a folk living in one spot and pursuing farming. The government had decided that their territory would be turned into a game preserve, and they would be given farm land deemed sufficient to support them. The transition was quite horrible for some time. Part of this horror was due to the government preventing them from hunting in their old territory. In addition, the nomadic ethical system simply did not function for them as farmers and caused them to make decisions that made their situation worse. Before they learned the trader ethic they very nearly died out. Today they live very much as their farmer neighbors do, and are surviving, even thriving to a modest degree. They raise crops and livestock. They even have a small surplus which lets them trade, and they are begining to do such things as send some of their children to college.
Jacobs makes the case that humans have two fundamental ways of getting a living: guardian and trader. In contrast, other animals only have one way.
So it is not specifically money, but commerce (or trade), that is characteristic of humans. And it is not instead of but in addition to the guardian system. And it expresses itself through there being two distinct systems of ethics.