Reputation (or "Gift") economies are fascinating things, and a neat look into pre-(and maybe post-?)money society.
In some ways, the analogy to currency holds: If you do a favor for me, I'm more likely to do a favor for you, there's a baked in "inflation" in that reputation decays over time. But in a lot of important ways, it differs: Me doing a favor for someone actually can make me more likely to do a future favor for them, versus no exchange taking place at all, depending on our level of rapport and the size of the favor. Reputation also rewards things that no-one would bother to quantify under currency economies: small favors, being enjoyable company, loyalty. Reputation is also aggressively local: just because Alice thinks very highly of me, it doesn't mean that Bob is more likely to help me, unless he and Alice have some kind of relationship, in which case I'd be leveraging not only what little rapport I have with Bob, but also my rapport with Alice, and Alice's rapport with Bob. This is a main weakness of such systems, because currency lets you deal with people you've never met before just as easily as your local barber. That leads to the necessity of strong hospitality cultures, because outsiders otherwise have no access to resources at all.
From a modeling perspective, currency lives in the nodes of a graph: Alice has some money, Bob has some money, I have some money. Reputation, in contrast, lives on the edges: I have rapport with Bob and Alice, Bob has rapport with Alice and I, etc.
On tangentially related note, it's been a bug-bear of mine that primary education history lessons always present barter as the premonetary economic system. This is absurd on its face: if Bob is a brilliant cobbler, and I'm a farmer with shoes that regularly need repairs, I'm not going to break down into whole negotiation of how many bushels of wheat or worse yet something more perishable like vegetables from my garden a new shoe sole costs, and how I'm going to save up and store enough for when my shoes need a fix. I'm going to build a reputation with Bob, by sharing with him excess from my garden, and in exchange he'll be willing to fix up my shoes when needed. Straight-up "I'll give you 4 chickens for a coat" barter is very rarely observed in societies before the introduction of currency.