I'm also getting same error, I have followed the code from web-socket-cookbook-example.
also added for better error fetching:

    try {
      final channel = WebSocketChannel.connect(
        Uri.parse('ws://nixbin:3000'),
      );

      channel.stream.listen(
        (message) => print('Received: $message'),
        onError: (error) => print('Error: $error'),
        onDone: () => print('Done'),
      );
    } catch (e) {
      print('Connection error: $e');
    }

nixbin is host only accessible within my devices.
getting error

A Dart VM Service on sdk gphone64 x86 64 is available at: http://127.0.0.1:42239/JIgF40mnNsc=/
E/flutter ( 7409): [ERROR:flutter/runtime/dart_vm_initializer.cc(41)] Unhandled Exception: WebSocketChannelException: Instance of 'WebSocketException'
E/flutter ( 7409): 
The Flutter DevTools debugger and profiler on sdk gphone64 x86 64 is available at:
http://127.0.0.1:9101?uri=http://127.0.0.1:42239/JIgF40mnNsc=/
I/flutter ( 7409): Error: WebSocketChannelException: Instance of 'WebSocketException'
I/flutter ( 7409): Done
E/flutter ( 7409): [ERROR:flutter/runtime/dart_vm_initializer.cc(41)] Unhandled Exception: WebSocketChannelException: Instance of 'WebSocketException'
E/flutter ( 7409): 
D/ProfileInstaller( 7409): Installing profile for com.example.project01

Is there anything missing? is it a bug for self-hosted websockets.

I am also experiencing this issue

So I did some digging, and it appears that the exception caught here is indeed of type 'WebSocketChannelException', looking into its code it appears to have 2 exposed variables:

1. message
2. inner

As many of us have tried, both of these return "Instance of WebSocketException", which is indeed not what we want. However, if we look into that class' code, it also has the same 2 exposed variables. These are what we want!

In order to get the error message itself, we'll need to access the WebSocketChannelException's inner's message. So in @fulstadev case, it'll look like this:
e.inner.message

Not sure if this is a bug or not since it seems weird that we'd have to access two identical objects just to get the error message, but doing the above did return the error message I've been looking for.

Pinging so maybe my solution above can help you too... @kirstenkoa @MROvaiz

Made it a new comment since I wasn't sure if editing a comment would actually notify you two.

I think that was fixed in #1756