It's all about the context of the relationship or conversation, Damien.
Weinberger focused on business as a conversation. Kiva is, in my humble opinion, primarily a social conversation. While money does exchange hands and drive entrepreneurship, it's ultimately not about the money.
Take Ariel's experiment involving attorneys and financially poor elderly. When attorneys were approached to donate their services at no cost to elderly individuals who could not otherwise afford representation, a strong majority of the attorneys agreed to provide help. By eliminating financial gain from the equation, Ariely was able to shift the conversation - the relationship - to a social context. That is to say, a conversation about people helping people.
The experiment was run again. This time, attorneys were asked to provide their services at a substantial discount from their regular fees. One would think that, perhaps, more attorneys might be interested in providing help. After all, they would receive some money rather than none at all. However, it turns out we are not the most rational beings. The number of attorneys willing to provide help dropped dramatically to a minority. Ariely attributed the drop to a shift from a social norm to a "business norm". In a nutshell, Ariely believes that the introduction of financial gain into the conversation caused attorneys to compare what they could earn per hour, with the rate they were offered - a business conversation. Why earn, say, $75 per hour when you can earn $150 per hour?
From a business perspective, it's a rational response until we consider that a majority of the attorneys were willing to give the same services away for free in the earlier experiment.
Why provide this example? Because I think it gets to the heart of Kiva. Kiva is a not-for-profit peer-to-peer lender. By definition members are not in it for profit. It's not a business conversation.
Great example.

I repeat myself, just to clarify. I wasn't making a point on Kiva being merely a market, I thought
you instead were thinking in terms of
markets at the beginning as mentioning personal choice at the heart of Kiva and thus felt to say something about thinking in terms of "individual choice only" might have sounded, but I misjudged I think.

One thing more to clarify: I didn't say the word "
business", but the word "
market". It's a very different thing. Market is about exchange, the process,
mechanism , not necessarily money. (although money is the most commonly recognized form of exchange, but still not the only one)
Business is a different thing and regards the
purpose/goal and the words surely are not to be easily swapped.
In any case, I was only discussing the notion, and we could discuss more about its elements (supply, demand, exchange, social relation embedded),
not advocating for it.
Hope this is clear now.

D.