Yep, what you're referring to (the "Powered by <browser>" screen) is done with the Custom Tabs API. Some folks might also refer to it as the in-app browser. Any browser can implement a CustomTabsService and the whole feature respects the default browser setting.
The Fastmail app is actually a good example of something that uses the WebView APIs though. Fastmail's UI is entirely web-based and their whole app is rendered inside of a WebView, provided by the Chromium-based "Android System WebView" app. If I remember correctly, the list of allowed signatures for a WebView provider is baked into the OS, so no third party can provide an implementation without root access.
The Fastmail app is actually a good example of something that uses the WebView APIs though. Fastmail's UI is entirely web-based and their whole app is rendered inside of a WebView, provided by the Chromium-based "Android System WebView" app. If I remember correctly, the list of allowed signatures for a WebView provider is baked into the OS, so no third party can provide an implementation without root access.