Google Play is not included with some versions of Console OS, including open-source builds.
You can, however, still use most Google services with Console OS builds that lack Google Play. To do this, however, you will need to disregard Google 2-factor authentication instructions for typical Android devices. Google's 2-factor instructions assume that you have Google Play installed on your Android.
Instead, you should follow steps for an Other device, in Google 2-factor speak. This includes setting an app-specific password. You will need to set an app-specific password for each service separately.
Using an app-specific password, you can define Gmail to work with the Email app, for example.
Google has a guide for setting up app passwords that you can follow for this process.