After reading this comment I tried to disable the 2FA with phone apps that I never asked for. Curiously, I'm not even signed in the app - I'm signed in Google Calendar but I get the prompts in Gmail app where I'm signed into work account only.

Anyway, it's not possible to configure 2FA in the way we want. The Google prompts configuration says "To turn off Google prompts on a device, sign out of your Google Account on that device."

There's no way to enforce the Authenticator. Not even make it default.