Recently most Google apps have been ported to Material Design 2.
See the new Android Messages redesign.
I would like to use this style in my app; in particular, I really like the new top bar in the home page, the "verbose" floating action button with text and the new font.
Is it possible to include it in an app or is it only for apps made by Google?
Summary of differences is here
https://9to5google.com/2018/04/26/what-is-material-design-2-examples-launch-io/
The only thing you will NOT BE ABLE TO USE is Google's Product Sans font as its closed source. But you should be able to implement everything else however.
Yes you can include it in your app. you just need to add the following dependency in your app:
implementation 'com.android.support:design:28.0.0-rc01'
You can check out some cool components like Gmail-like chip view, bottom app bar, etc. here: https://material.io/develop/android/components/bottom-app-bar/
or you can watch this video: https://www.youtube.com/watch?v=SdoddLrywZU
Although it's in kotlin it might serve your purspose.