NTP: Prevent data loss in topSites after topSites update

[cezaraugusto]

Reviewers: Please focus review on commit ce9e621 since it's a fix for the reason the first commit 3203693 was reverted.

Fix brave/brave-browser#8781

Top sites (now called grid sites) now have their own storage. This change allows the migration process from a shared to a dedicated storage.

This PR also re-introduces 3203693 which was reverted in #4996

Fix brave/brave-browser#2971

Submitter Checklist:

• Submitted a ticket for my issue if one did not already exist.
• Used Github auto-closing keywords in the commit message.
• Added/updated tests for this change (for new code or code which already has tests).
• Verified that these changes build without errors on
  • Android
  • iOS
  • Linux
  • macOS
  • Windows
• Verified that these changes pass automated tests (unit, browser, security tests) on
  • iOS
  • Linux
  • macOS
  • Windows
• Verified that all lint errors/warnings are resolved (npm run lint)
• Ran git rebase master (if needed).
• Ran git rebase -i to squash commits (if needed).
• Tagged reviewers and labelled the pull request as needed.
• Request a security/privacy review as needed.
• Add appropriate QA labels (QA/Yes or QA/No) to include the closed issue in milestone
• Public documentation has been updated as necessary. For instance:
  • https://github.com/brave/brave-browser/wiki/Deviations-from-Chromium-(features-we-disable-or-remove)
  • https://github.com/brave/brave-browser/wiki/Proxy-redirected-URLs
  • https://github.com/brave/brave-browser/wiki/Fingerprinting-Protection-Mode
  • https://github.com/brave/brave-browser/wiki/Brave%E2%80%99s-Use-of-Referral-Codes
  • https://github.com/brave/brave-browser/wiki/Custom-Headers
  • https://github.com/brave/brave-browser/wiki/Web-compatibility-issues-with-tracking-protection
  • https://github.com/brave/brave-browser/wiki/QA-Guide

Test Plan:

Existing profile

1. Before checking out this PR, keep in mind your pinned top sites; if you have none, pin some tiles and keep in mind their URL and grid position.
2. Check out this PR. Assert your pinned top sites are there, on the same position as before.

Clean profile

1. Top sites should work as expected. As you visit new sites, grid should populate with them.
2. Pinning/reordering/excluding should work as expected.

Reviewer Checklist:

• New files have MPL-2.0 license header.
• Request a security/privacy review as needed.
• Adequate test coverage exists to prevent regressions
• Verify test plan is specified in PR before merging to source

After-merge Checklist:

• The associated issue milestone is set to the smallest version that the
  changes has landed on.
• All relevant documentation has been updated.

[cezaraugusto]

Added some more tests cc @bsclifton

[bsclifton]

Manually tested - works great! Code is nice too- thanks for adding tests 😄👍

Added @simonhong to review as he's working on the Super Referral project which also uses top site tiles 😄

[bsclifton]

Reviewed w/ @simonhong - changes shouldn't cause a problem w/ spec as the color for top site tiles is mobile only I believe. Ready to merge! 😄