You can definitely write articles about the Kindle, but you shouldn’t use the definite article with (the) Kindle. That’s the thrust of a Wall Street Journal article that discusses how careful representatives of many modern companies are to avoid using “a” or “the” with their products. Kindle, Nook, iPhone, BlackBerry, Wii, the list goes on.
The idea seems to be to make the devices seem “personal and human”, presumably by talking about them more as if they were people than products. (After all, you wouldn’t say “the Chris Meadows”. Well, unless you were asking, “You’re Chris Meadows? The Chris Meadows?” Which I can’t say has happened to me yet, though it actually has happened to a friend of mine.)
Still, it also can make it seem like these marketroids are somewhat more detached from reality than everyday people, who use articles as a matter of course. And it seems prone to teach impressionable youths the wrong grammatical lessons.
The whole thing seems more than a little silly. Whether it’s the Kindle or just Kindle, the important thing is how many people use it to read, not how you talk about it.