If they don't like an app, it doesn't get in the store, and, unless you've broken your device (which is of questionable legality), you're limited to Apple approved apps.
You're limited to Apple-approved native apps. You can "install" HTML5-based apps (including ones with local storage that work offline) without any approval process from Apple. You won't have full access to all the device capabilities, but there you go.
Besides, GP was using examples of media content, not application content. Apple doesn't restrict what content you can view or load, and the formats they support are open standards. Hardly a restrictive regime, App Store aside.
Amazon can delete books you've bought, or even edit the content of books without your consent.
Again, when we're talking about device capabilities, that's a tiny slice. The Kindle supports a great many things that don't have Amazon DRM, and Amazon does not have the capabilities you describe with media that's not DRM-controlled. Again, you can read whatever you want on a Kindle provided it's in one of a handful of formats.
The GP is arguing that Google restricts what you can stream -- that is, content -- to the Chromecast, and likens it to Apple and Amazon. Firstly, it isn't true - you can stream pretty much any content you want to a Chromecast, as long as it'll load in a Chrome tab without 3rd-party plugins (and that means common open standards as well as Flash content). Secondly, even if it were true that Google makes content or content-provider restrictions, that would make them exceptional in the industry. Neither Apple nor Amazon prevent you from viewing content or accessing information you choose, so long as it's in one of the ubiquitous and standard formats.