[web] Style the AppSwitcher

Summary:
Style the AppSwitcher according to the new designs (minor changes): https://www.figma.com/file/a1nkbWgbgjRlrOY9LVurTz/Comm-%2F-Desktop-app?node-id=7780%3A136918&t=zg5f0X86eUyYnqI7-4
The badge on the Chat tab is supposed to stay as it was - in the corner of the icon.

Test Plan: Run web app, checked that the component displays correctly

Reviewers: michal, kamil, bartek, kuba

Reviewed By: kamil

Subscribers: ashoat, tomek, atul

Differential Revision: https://phab.comm.dev/D6849