Repository: flutter
Issue: Manually writing the Arabic date in the datePicker is not working
Labels: f: material design,f: date/time picker,a: internationalization,has reproducible steps,P2,team-design,triaged-design,found in release: 3.16,found in release: 3.17

### Is there an existing issue for this?

- [X] I have searched the [existing issues](https://github.com/flutter/flutter/issues)
- [X] I have read the [guide to filing a bug](https://flutter.dev/docs/resources/bug-reports)

### Steps to reproduce

1. Call showDatePicker with locale set to 'ar'
2.  Switch to input field
3. Write the date manually in Arabic letters

### Expected results

I expect to that when entering the date in the input field in Arabic it will accept it.

### Actual results

The actual is it shows an error saying "تنسيق غير صالح" , which something like "Invalid format".
Note that choosing a date from the picker then switching back to input field mode will accept the value which is really strange.
There's no difference between these two values:
- ١٦‏/١١‏/٢٠٢٣ (Accepted)
-  ٢٠٢٣/١١/١٦ (Not accepted)

Also, It would be great if English numbers (1,2,3,4,..) were allowed, since most keyboards don't have the Arabic numbers.

### Code sample

<details><summary>Code sample</summary>

```dart
import 'package:flutter/material.dart';
import 'package:flutter_localizations/flutter_localizations.dart';

void main() {
  runApp(const MyApp());
}

class MyApp extends StatelessWidget {
  const MyApp({super.key});

  @override
  Widget build(BuildContext context) {
    return MaterialApp(
      debugShowCheckedModeBanner: false,
      localizationsDelegates: const [
        GlobalMaterialLocalizations.delegate,
        GlobalWidgetsLocalizations.delegate,
        GlobalCupertinoLocalizations.delegate,
      ],
      supportedLocales: const [Locale('ar')],
      theme: ThemeData(
        colorScheme: ColorScheme.fromSeed(seedColor: Colors.deepPurple),
        useMaterial3: true,
      ),
      home: const HomePage(),
    );
  }
}

class HomePage extends StatelessWidget {
  const HomePage({super.key});

  @override
  Widget build(BuildContext context) {
    return Scaffold(
      body: Center(
        child: ElevatedButton(
            onPressed: () => showDatePicker(
                  context: context,
                  locale: const Locale('ar'),
                  firstDate: DateTime(2018),
                  lastDate: DateTime.now(),
                ),
            child: const Text('Click me')),
      ),
    );
  }
}

```

</details>


### Screenshots or Video

<details>
<summary>Screenshots / Video demonstration</summary>

[[Upload media here]](https://github.com/flutter/flutter/assets/57685540/3925e009-3d71-4464-abb7-c06ea7287ef9)

</details>





### Logs

<details><summary>Logs</summary>

```console
[Paste your logs here]
```

</details>


### Flutter Doctor output

<details><summary>Doctor output</summary>

```console
[!] Flutter (Channel stable, 3.16.0, on Linux Mint 21.2 5.15.0-88-generic,
    locale en_US.UTF-8)
    • Flutter version 3.16.0 on channel stable at
      /home/hasan/development/flutter
    ! Warning: `dart` on your path resolves to
      /home/linuxbrew/.linuxbrew/Cellar/dart/3.2.0/libexec/bin/dart, which is
      not inside your current Flutter SDK checkout at
      /home/hasan/development/flutter. Consider adding
      /home/hasan/development/flutter/bin to the front of your path.
    • Upstream repository https://github.com/flutter/flutter.git
    • Framework revision db7ef5bf9f (2 days ago), 2023-11-15 11:25:44 -0800
    • Engine revision 74d16627b9
    • Dart version 3.2.0
    • DevTools version 2.28.2
    • If those were intentional, you can disregard the above warnings; however
      it is recommended to use "git" directly to perform update checks and
      upgrades.

[✓] Android toolchain - develop for Android devices (Android SDK version 33.0.2)
    • Android SDK at /home/hasan/Android/Sdk
    • Platform android-34, build-tools 33.0.2
    • Java binary at: /home/hasan/development/android-studio/jbr/bin/java
    • Java version OpenJDK Runtime Environment (build
      11.0.15+0-b2043.56-8887301)
    • All Android licenses accepted.

[✓] Chrome - develop for the web
    • Chrome at google-chrome

[✓] Linux toolchain - develop for Linux desktop
    • Ubuntu clang version 14.0.0-1ubuntu1.1
    • cmake version 3.22.1
    • ninja version 1.10.1
    • pkg-config version 0.29.2

[✓] Android Studio (version 2022.1)
    • Android Studio at /home/hasan/development/android-studio
    • Flutter plugin can be installed from:
      🔨 https://plugins.jetbrains.com/plugin/9212-flutter
    • Dart plugin can be installed from:
      🔨 https://plugins.jetbrains.com/plugin/6351-dart
    • Java version OpenJDK Runtime Environment (build
      11.0.15+0-b2043.56-8887301)

[✓] VS Code (version 1.84.2)
    • VS Code at /usr/share/code
    • Flutter extension version 3.76.0

[✓] Connected device (2 available)
    • Linux (desktop) • linux  • linux-x64      • Linux Mint 21.2
      5.15.0-88-generic
    • Chrome (web)    • chrome • web-javascript • Google Chrome 119.0.6045.159

[✓] Network resources
    • All expected network resources are available.

! Doctor found issues in 1 category.

```

</details>