This is a simplistic view over money and currencies in general. Money is a form of credit. It's an abstraction in the sense that it unifies preferences and signals under a single dimension. It's an approximation in the sense that, when you buy something for 20 bucks, it's not that the something is actually worth 20 bucks, but that you value 20 bucks (and, by extension, the other things you could buy with them) than the something.

In this regard, while it's kinda wrong to say "it costs money to care for people", caring for people is not free either - you have to spend something: time, labor, psyche, favors, etc. All of these can be acquired or replaced by trading with others (which is preferable to do through money): buying tools/machines, hiring workers/services, acquiring means to connect with others, and so on. Turns out money is just a more verifiable and simplified form of favors.