They really are a type of currency, people don't think of them as such because they aren't freely convertable, you're pretty much stuck using them on airfare or some fairly limited catalog of products, but companies (like credit card companies) buy them from airlines to give out as rewards.

Amazon could start a new blockchain then give the coins out as rewards points.  You could then redeem them for money off on your purchase. They could also sell them to other web sites, you buy an airline ticket and get X amazonRewardPoints.  Since it is based on the blockchain, any partner could verify the number of points out there.  Also, they could accept points knowing they were free to re-distribute them and that Amazon was effectively fixing their value.  Pretty soon you'd have a large network of merchants both giving out and accepting points without having to trust each other not to create an arbitrary amount of points.