Fix #8095: Favorites are shown in Private Tab mode when switching the apps

[soner-yuksel]

Summary of Changes

This pull request fixes #8095

Hide Active background favorites when backgrounding the app in private mode

Submitter Checklist:

• Unit Tests are updated to cover new or changed functionality
• User-facing strings use NSLocalizableString()
• New or updated UI has been tested across:
  • Light & dark mode
  • Different size classes (iPhone, landscape, iPad)
  • Different dynamic type sizes

Test Plan:

• Begin Brave
• Open NTP in Private Tab mode
• Tap Search button > Use the app switch gesture to see all opened apps > Observe

Screenshots:

Reviewer Checklist:

• Issues include necessary QA labels:
  • QA/(Yes|No)
  • bug / enhancement
• Necessary security reviews have taken place.
• Adequate unit test coverage exists to prevent regressions.
• Adequate test plan exists for QA to validate (if applicable).
• Issue and pull request is assigned to a milestone (should happen at merge time).