How can I allow my friend to see my screen using a web browser without requiring installation of any app on their or my device?
The W3C spec that covers screen capture for the Media API: 

https://www.w3.org/TR/screen-capture/ 

is in Working Draft and as of May 2020 does appear to have some support on Desktop browsers but is not supported by any of the main Mobile browsers:

https://developer.mozilla.org/en-US/docs/Web/API/Screen_Capture_API