Example: After opening Stack Overflow, you will be greeted with Google's Single Sign-On dialog pop-up ("Use your Google Account to sign in to <the website's name>"), which covers a significant amount of space of the website.

This is prominent especially on a small smartphone screen, where the pop-up covers about 40% of the screen space, so you are indirectly forced to interact with the pop-up to close it.

Is it a deliberate intention of Google to force you to interact with the pop-up so that it can gain additional privileges inside the browser to track you better?

AFAIK, browsers require user interaction to play sound. Are there other APIs which require prior user interaction to allow access to them? Could such user-interaction-requiring-APIs allow Google to get a better fingerprint of your browser?