Amazon will sell as many Kindles as they make, but the number they will make will not be the most they could make.
The logic of selling a product which has a profit model unrelated to the cost of goods sold is tricky. The incentives are different. The risk is not selling too few but selling too many.
For this reason I think the total number of new Fire units to be sold has already been determined by a production order. You cannot think about the business from a demand point of view. If the product would be free then the demand would be infinite. The decision about how many will be “sold” will depend on the goodwill of the producer.
What Amazon tries to do with the brand is ensure that the Fire is in the hands of its most ravenous consumers. That is why it’s not sold in all markets or through all channels. They are sold through Amazon.com in the US (limited sales in UK as well). This is because a large number of the product in the hands of users who only use it for browsing or in areas where Amazon does not have content deals or where its ads are poorly targeted (e.g. India, Indonesia or Madagascar) would be a disaster. It may not be all that helpful to Google to have Android in those markets but as you would expect it’s still a profitable business for Apple.
So one way to think of the Fire is as a promotional item (aka swag) for another business (Amazon.com). Using this frame of mind, assessing its “threat” to another business which charges for the product itself is like assessing whether free t-shirts from trade shows affect the sales of clothing or apparel in general. They do, but mostly the sale of cheap t-shirts. I doubt that people stop buying more functional clothing because they have hundreds of free t-shirts. And then there’s the problem of looking like an advertisement.