To answer your first question, AFAIK, standalone services that can be used on their own are prefixed with Amazon (eg S3, EC2, DDB), whereas services that are deeply integrated into AWS ecosystem are prefixed with AWS.
Disclaimer: I work for AWS, but this is not an official answer. What I said is correct to my best knowledge, but I cannot guarantee its correctness/accuracy.
1. When are services branded as AWS (AWS Fargate) vs Amazon (Amazon DynamoDB)?
2. Is BottleRocket a nod to SkyRocket [0] or a movie of the same name?
3. Why is it called Fargate [1]?
[0] https://en.wikipedia.org/wiki/Skyrocket
[1] https://youtube.com/watch?v=ye3-gUwu9tI&t=44m28s