Just a nitpick. "Fiduciary" doesn't mean "money", it means an entity which is legally bound to the best interests of the other party. Non-profit boards and board members have fiduciary duties.
However, to nitpick your nitpick: for non-profits there might be no other party - just the mission. Imagine a non-profit whose mission is to preserve the history and practice of making 17th-century ivory cuff links. It's just the organisation and the mission; sometimes the mission is for the benefit of another party (or all of humanity).