[web] Refactor search styling to match Figma

Summary:
Styling changes for Search making the component similar to the one from Figma in Members modal.

Figma (in members modal):

Implementation (now only in thread search)
Before:

After
v2

Test Plan: The thread search bar should now be similar to the search in modal design.

Reviewers: palys-swm, benschac, atul

Reviewed By: palys-swm, benschac

Subscribers: ashoat, Adrian, atul, karol-bisztyga

Differential Revision: https://phabricator.ashoat.com/D3368