It's an (albeit extremely large and diverse) e-commerce website, not a platform. What software do you build that runs on top of Amazon?
Ignoring AWS, there are lots of complex systems you can build on top of Amazon. One example is Fulfillment by Amazon. You can integrate this service with your ecommerce website, and Amazon can warehouse your inventory in its facilities nationwide, charging you based on the volume of your inventory. They will automatically select a warehouse closest to your clients that has your item in stock and pack and ship it, and even order more inventory from your supplier when a certain warehouse inventory level falls below a certain threshold. It is basically S3 backed CloudFront for physical goods.
From the article:
A software platform is truly a foundation on which entire businesses can be built. It encompasses not just a technical infrastructure but also a user experience framework, usually some form of a selling channel, and a defined large-scale developer ecosystem
If that isn't the very definition of Amazon, I don't know what is. In theory I could build an entire business on top of Amazon's infrastructure (and not just a software business). I could have outsourced labor write or edit content using Mechanical Turk. I could publish it using Amazon's self-publishing tools. I could sell it through Amazon's marketplace or on an ecommerce site hosted on AWS. I could collect payments using Amazon FPS. I can ship products using Fulfillment by Amazon. You most certainly can build a business using someone else's warehouse, just as Heroku and RightScale can build businesses on someone else's datacenter. Fulfillment is a service, but is is a PaaS - a robust logistics platform as a service.
Windows is a platform.
Heroku is a platform.
Facebook is a platform (barely).
Android is a platform.
These are things you write software for that sits on top of their existing stack. Something providing webservices, no matter how many there are or how much business they generate, is not a platform.