Repository: flutter
Issue: [mobile] video_player accidentally allows attaching one VideoPlayerController to more than one VideoPlayers
Labels: p: video_player,package,team-ecosystem,has reproducible steps,P2,found in release: 3.3,found in release: 3.7,triaged-ecosystem

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## Steps to Reproduce

<!-- Please include full steps to reproduce so that we can reproduce the problem. -->

- Modified the `_BumbleBeeRemoteVideoState` code in `flutter_plugins/packages/video_player/video_player/example/lib/main.dart` with sample code
- Execute `flutter run -d Chrome` 

**Expected results:** <!-- what did you want to see? -->

Two videos play synchronously with no problem.

**Actual results:** <!-- what did you see? -->

Only the second video plays normally.

<details>
<summary>Code sample</summary>


```dart
class _BumbleBeeRemoteVideoState extends State<_BumbleBeeRemoteVideo> {
  late VideoPlayerController _controller;

  Future<ClosedCaptionFile> _loadCaptions() async {
    final String fileContents = await DefaultAssetBundle.of(context)
        .loadString('assets/bumble_bee_captions.vtt');
    return WebVTTCaptionFile(
        fileContents); // For vtt files, use WebVTTCaptionFile
  }

  @override
  void initState() {
    super.initState();
    _controller = VideoPlayerController.network(
      'https://flutter.github.io/assets-for-api-docs/assets/videos/bee.mp4',
      closedCaptionFile: _loadCaptions(),
      videoPlayerOptions: VideoPlayerOptions(mixWithOthers: true),
    );

    _controller.addListener(() {
      setState(() {});
    });
    _controller.setLooping(true);
    _controller.initialize();
  }

  @override
  void dispose() {
    _controller.dispose();
    super.dispose();
  }

  @override
  Widget build(BuildContext context) {
    return SingleChildScrollView(
      child: Column(
        children: <Widget>[
          Container(padding: const EdgeInsets.only(top: 20.0)),
          const Text('With remote mp4'),
          Container(
            padding: const EdgeInsets.all(20),
            child: AspectRatio(
              aspectRatio: _controller.value.aspectRatio,
              child: Stack(
                alignment: Alignment.bottomCenter,
                children: <Widget>[
                  VideoPlayer(_controller),
                  ClosedCaption(text: _controller.value.caption.text),
                  _ControlsOverlay(controller: _controller),
                  VideoProgressIndicator(_controller, allowScrubbing: true),
                ],
              ),
            ),
          ),
          Container(padding: const EdgeInsets.only(top: 20.0)),
          const Text('With remote mp4'),
          Container(
            padding: const EdgeInsets.all(20),
            child: AspectRatio(
              aspectRatio: _controller.value.aspectRatio,
              child: Stack(
                alignment: Alignment.bottomCenter,
                children: <Widget>[
                  VideoPlayer(_controller),
                  ClosedCaption(text: _controller.value.caption.text),
                  _ControlsOverlay(controller: _controller),
                  VideoProgressIndicator(_controller, allowScrubbing: true),
                ],
              ),
            ),
          ),
        ],
      ),
    );
  }
}
```

</details>

I think it's related to `buildView` implementation:

```dart
  @override
  Widget buildView(int textureId) {
    return HtmlElementView(viewType: 'videoPlayer-$textureId');
  }
```

~`HtmlElementView.viewType` requires unique as the comment said: `/// The unique identifier for the HTML view type to be embedded by this widget.`~
~However `Texture` used by iOS/Android doesn't have this restriction.~
Seems this is not the real reason after some trials.