Kindle Fire IS a true Android tablet, companies are free to modify Android how they want. And you can install the Google Play store on it if you want to.
Google doesn't endorse the Kindle Fire as a true Android tablet, and Amazon removed the Google Play Store so they could push their own app store. If it was a true Android tablet, it would give us the complete Android experience, Google and all. Amazon wants the Fire to be a gateway to their ecosystem, which is why they removed native access to the Google Play Store. It probably would have been easier for Amazon to design their own OS.
Android is just the backbone of the Kindle Fire's OS. It's not the core feature of the product.
Look at the Kindle Fire, then look at the Google Nexus 7, Samsung Galaxy Tab, etc. Does a Kindle give you the same experience as those other tablets that run Android natively? No. Amazon's tablet is nothing like a real Android tablet, because Amazon doesn't want it to be one.
If I took Windows, stripped it down, redesigned it into something completely different, and repackaged it into a PC, would that device still be a Windows PC?