[web] intorduce user profile menu to user profile modal

Summary:
This diff here introduces the logic for the user profile menu into the user profile modal

Linear task: https://linear.app/comm/issue/ENG-5225/introduce-userprofilemenu

Depends on D9457

Test Plan: Confirmed that the user profile menu works as expected.

Reviewers: atul, inka

Reviewed By: atul

Subscribers: ashoat, tomek, wyilio

Differential Revision: https://phab.comm.dev/D9458