How to add Screen Recording permission to Google Chrome in Catalina?
It looks like as long as an app does not try to capture or record the screen, the app cannot be manually added to the "Screen Recording" list. For other permission types such as Full Disk Access, I have successfully dragged and dropped an application to add to the list. 

In that case, you can open Google Chrome, visit this site and click the "Get Started" button. I have tried it on MacOS Catalina 10.15.4 with Google Chrome 83.0.4103.61 (Official Build) (64-bit), and it worked. After pressing the "Get Started" button, Chrome requested permission to record screen, and then afterwards, I could go to settings->Security &amp; Privacy and then grant permission for Screen Recording. 

During my test, I already had the Security &amp; Privacy window open, and already clicked the Lock icon to make changes and entered my password. 

Afterwards, I have tested with Google meetings and confirmed that I can share my screen. I have also confirmed that when I deselect Chrome from the Screen Recording list, I can't share the screen.

Hope that works for you, too.