Bug 1882308 - Updated custom tab to support private mode

(cherry picked from commit de4214bda7dc5298098fc99e10ec5a077c29043d)

app/src/main/java/org/mozilla/fenix/HomeActivity.kt

@@ -879,8 +879,8 @@ open class HomeActivity : LocaleAwareAppCompatActivity(), NavHostActivity {
 
     private fun setupTheme() {
         themeManager = createThemeManager()
-        // ExternalAppBrowserActivity handles it's own theming as it can be customized.
-        if (this !is ExternalAppBrowserActivity) {
+        // ExternalAppBrowserActivity exclusively handles it's own theming unless in private mode.
+        if (this !is ExternalAppBrowserActivity || browsingModeManager.mode.isPrivate) {
             themeManager.setActivityTheme(this)
             themeManager.applyStatusBarTheme(this)
         }

app/src/main/java/org/mozilla/fenix/browser/BaseBrowserFragment.kt

@@ -266,8 +266,8 @@ abstract class BaseBrowserFragment :
         _binding = FragmentBrowserBinding.inflate(inflater, container, false)
 
         val activity = activity as HomeActivity
-        // ExternalAppBrowserActivity handles it's own theming as it can be customized.
-        if (activity !is ExternalAppBrowserActivity) {
+        // ExternalAppBrowserActivity exclusively handles it's own theming unless in private mode.
+        if (activity !is ExternalAppBrowserActivity || activity.browsingModeManager.mode.isPrivate) {
             activity.themeManager.applyStatusBarTheme(activity)
         }
 
@@ -1574,8 +1574,8 @@ abstract class BaseBrowserFragment :
             activity?.exitImmersiveMode()
             (view as? SwipeGestureLayout)?.isSwipeEnabled = true
             (activity as? HomeActivity)?.let { activity ->
-                // ExternalAppBrowserActivity handles it's own theming as it can be customized.
-                if (activity !is ExternalAppBrowserActivity) {
+                // ExternalAppBrowserActivity exclusively handles it's own theming unless in private mode.
+                if (activity !is ExternalAppBrowserActivity || activity.browsingModeManager.mode.isPrivate) {
                     activity.themeManager.applyStatusBarTheme(activity)
                 }
             }