..but why? How is this relevant to their business? I mean, I do understand that AWS and Amazon (the e-commerce) don't have anything in common, but why to focus on such a product that already exists (and that's even better)?
It's an addition to SNS, which is AWS's notification service -- you set alerts, and if a server goes down, or any other thing you set up a trigger for, it messages you. It's just expanding to new markets.
Their goal is the be the infrastructure of the internet. This is an infrastructure product. Yes there are competitors, but if you're already using other AWS services, there's a good chance you'll use their solution for any given problem.