Apparently it does. This is just bullshit. They didn't explain how they achieved feature parity - because they didn't. They just wanted to ride on the hate wave. "Weeee it's just an Android app on AOSP, just make an app loool" - sure, so show us it actually does the same thing.
You can't because the standard Android phone operating systems and Play store restrictions don't allow it. You need to build (slightly) customized AOSP and flash it to do the same thing. Or root the phone, but that's still very distant from "it could be just an app".
BTW I would love if there was a way to do it. I have my own humble voice assistant system and just installing it as an app on my phone to fully replace Google Assistant would be perfect. But you can't do that (and of course it's a total nonstarter on iOS), so it runs on a Raspberry Pi with standard Raspbian and the app package - practically the same solution.