Resolve SE order related issue

[deeppandya]

Resolves brave/brave-browser#18777

Submitter Checklist:

• [] I confirm that no security/privacy review is needed, or that I have requested one
• There is a ticket for my issue
• Used Github auto-closing keywords in the PR description above
• Wrote a good PR/commit description
• Added appropriate labels (QA/Yes or QA/No; release-notes/include or release-notes/exclude; OS/...) to the associated issue
• Checked the PR locally: npm run test -- brave_browser_tests, npm run test -- brave_unit_tests, npm run lint, npm run gn_check, npm run tslint
• Ran git rebase master (if needed)

Reviewer Checklist:

• A security review is not needed, or a link to one is included in the PR description
• New files have MPL-2.0 license header
• Adequate test coverage exists to prevent regressions
• Major classes, functions and non-trivial code blocks are well-commented
• Changes in component dependencies are properly reflected in gn
• Code follows the style guide
• Test plan is specified in PR before merging

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, 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-Protections
  • 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-Exceptions-in-Brave
  • https://github.com/brave/brave-browser/wiki/QA-Guide
  • https://github.com/brave/brave-browser/wiki/P3A

Test Plan:

[SergeyZhukovsky]

++

[kjozwiak]

Verification PASSED on Samsung S10+ running Android 11 using the following build:

1.33.36 Chromium: 95.0.4638.54

Test Case #1 (Locale Used: Poland)

• ensured that tapping on the URL bar on a fresh install displays the SE on-boarding modal
• ensured that selecting Brave as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting DDG as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Google as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Bing as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Qwant as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Startpage as the search engine correctly sets both Standard & Private via SE settings

Test Case #2 (Locale Used: Spain)

• ensured that tapping on the URL bar on a fresh install displays the SE on-boarding modal
• ensured that selecting Brave as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting DDG as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Qwant as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Bing as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Google as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Ecosia as the search engine correctly sets both Standard & Private via SE settings
• ensured that selecting Startpage as the search engine correctly sets both Standard & Private via SE settings

Test Case #3 (Locale Used: USA)

• ensured that there's no SE on-boarding modal when tapping on the URL bar on a fresh install
• ensured that Brave is set as the default browser for both Standard & Private via SE settings

Test Case #4 (Locale Used: France)

• ensured that there's no SE on-boarding modal when tapping on the URL bar on a fresh install
• ensured that Brave is set as the default browser for both Standard & Private via SE settings

[kjozwiak]

Adding missing 1.33.x - Nightly milestone 👍