You're operating from a premise that misunderstands the nature of money in a market. Money is just a measure of value, a metric of human endeavor, and so it's traded on that basis. Publicly traded for-profit corporations have a fiduciary duty to maximize shareholder value. It only happens that we measure value in, say, Dollars or Yen. We could measure it in ounces of gold or bushels of oranges. For-profit companies transcend currency, and could exist in a barter economy.

So it's not that money (as used to mean currency) is a defining characteristic of for-profit business, it's that value is. If a for-profit business existed to "maximize shareholder value in terms of cows", they would try to maximize shareholder NPV as measured in cows. The reason that for-profit companies exist is because the economics-theoretic ideal point for production in an economy (the point that benefits all consumers the most) is at supply-demand equilibrium, which is where profits are maximized for an individual company. The fact that there are non-profit companies is simply indicative of market inefficiency--in a "perfect" market, all companies would be for-profit. That doesn't mean that nonprofits shouldn't exist right now; in fact, it means that they have to.

So let's change "money" to "value" to be more specific. And then yes, value is indeed a defining characteristic of business, no matter how you measure it: currency accumulated via industry, children vaccinated, or political points spread. That's because humans tend to take action to maximize what they value most, humanitarian or self-serving, and business is the systematic application of human action.