Fix for issue 140372 (#144947)

*Continuing work from the PR https://github.com/flutter/flutter/pull/140373*

Add the ability to set route settings on PopupMenuButton

Fixes https://github.com/flutter/flutter/issues/140372

Added UTs as requested

packages/flutter/lib/src/material/popup_menu.dart

@@ -1150,6 +1150,7 @@ class PopupMenuButton<T> extends StatefulWidget {
     this.clipBehavior = Clip.none,
     this.useRootNavigator = false,
     this.popUpAnimationStyle,
+    this.routeSettings,
     this.style,
   }) : assert(
          !(child != null && icon != null),
@@ -1345,6 +1346,11 @@ class PopupMenuButton<T> extends StatefulWidget {
   /// If this is null, then the default animation will be used.
   final AnimationStyle? popUpAnimationStyle;
 
+  /// Optional route settings for the menu.
+  ///
+  /// See [RouteSettings] for details.
+  final RouteSettings? routeSettings;
+
   /// Customizes this icon button's appearance.
   ///
   /// The [style] is only used for Material 3 [IconButton]s. If [ThemeData.useMaterial3]
@@ -1412,6 +1418,7 @@ class PopupMenuButtonState<T> extends State<PopupMenuButton<T>> {
         clipBehavior: widget.clipBehavior,
         useRootNavigator: widget.useRootNavigator,
         popUpAnimationStyle: widget.popUpAnimationStyle,
+        routeSettings: widget.routeSettings,
       )
       .then<void>((T? newValue) {
         if (!mounted) {

packages/flutter/test/material/popup_menu_test.dart

@@ -926,6 +926,39 @@ void main() {
     expect(tester.getTopLeft(popupFinder), buttonTopLeft);
   });
 
+  testWidgets('Popup menu with RouteSettings', (WidgetTester tester) async {
+    final Key buttonKey = UniqueKey();
+    const RouteSettings popupRoute = RouteSettings(name: '/popup');
+    late RouteSettings currentRouteSetting;
+
+    await tester.pumpWidget(
+      MaterialApp(
+        navigatorObservers: <NavigatorObserver>[
+          _ClosureNavigatorObserver(onDidChange: (Route<dynamic> newRoute) {
+            currentRouteSetting = newRoute.settings;
+          }),
+        ],
+        home: Scaffold(
+         body: PopupMenuButton<int>(
+          key: buttonKey,
+          routeSettings: popupRoute,
+          itemBuilder: (_) => <PopupMenuItem<int>>[
+            const PopupMenuItem<int>(value: 1, child: Text('Item 1')),
+            const PopupMenuItem<int>(value: 2, child: Text('Item 2')),
+          ],
+          child: const Text('Show Menu'),
+          ),
+        ),
+      ),
+    );
+
+    final Finder buttonFinder = find.byKey(buttonKey);
+    await tester.tap(buttonFinder);
+    await tester.pumpAndSettle();
+
+    expect(currentRouteSetting, popupRoute);
+  });
+
   testWidgets('PopupMenu positioning around display features', (WidgetTester tester) async {
     final Key buttonKey = UniqueKey();