[Skills] Add custom emoji picker component

Adds the skills-emoji-picker Lit component, including its styles,
template, and implementation. Includes the emoji category data and a
dedicated test suite.

Demo: https://screencast.googleplex.com/cast/NTc1OTg3MzE1NjM4MjcyMHxlMmEyMDVmMS1mMA
Bug: 483955922
Change-Id: Idb26f405d8129775a09db6b36eaa57feb07a2edb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/7634961
Reviewed-by: Ian Vollick <vollick@chromium.org>
Reviewed-by: Rebekah Potter <rbpotter@chromium.org>
Reviewed-by: Christine Ying <chrstne@google.com>
Commit-Queue: Howard Chang <howardcha@google.com>
Cr-Commit-Position: refs/heads/main@{#1602108}