When you are connecting signals to functions, it is possible to add one or more custom parameters to the signal handler. As seen earlier, connect_object() can be used to pass a particular widget into a function, however an alternative is to use connect() with extra parameters for the information you wish to use inside the function.
The handler function would then need to have accepted three parameters - the event, the GtkWindow object that emitted the signal, and the custom parameter $menu. The only difference here is that behind the scenes a little more data needs to be passed around for the handler function to be called with the extra parameter.
Here is a complete code example you can try out to get the idea: