Being tangentially involved in this project it was my understanding that the devices were financed in large part by the marketing departments of the brands that had buttons, which is why they were tied to brands.
"We lose a little money on every customer, but we make it up on volume." :-)
I think in Amazon's case it was just experimentation with a sunk cost. Based on the discussion above it looks like it's impossible to form a company to sell these to turn a profit, at least in the dash button form.
The original purpose of the “hackable” button was to give a fun piece of experimentation kit that worked with AWS and encouraged people to try it out. I set up my AWS account just to use one.
Was the original/intended purpose to always sell them at cost and make the profits up through increased sales?