I don't believe it does give them your credentials? It's more like running something like AutoHotkey and having it send screenshots to Perplexity and inject whatever input events they send back, but integrated directly into the browser instead of running as a separate program.

They are not interacting with Amazon at all. They are sending control commands to your computer, where you've opened a session logged into Amazon as you. Amazon is claiming they can't run commands on your computer.

The whole thing becomes entirely absurd if you consider that a cloud model isn't even fundamental to the setup; you could just as well slot in a local model or codex or anything else and the scenario would be exactly the same. Perplexity is just acting as a model provider here.