How to wait for an action of another device like Google Prompt?

Using Google's two-factor authentication, as you try to login, a prompt is pushed to your phone so you just have to acknowledge with a tap that you are the one signing in.

I was wondering how to wait for an action sent to another device using http.

How is the prompt pushed to your phone? Is it a classic Android notification?

How does the website wait for the acknowledgment? I noticed that while waiting, a xhr request is pending for a maximum of 20 seconds. What is the cost of a pending request?

What does this request do server side? Does it periodically ask if the authentication done? Is it done in a stateless way?