Wpf- Unable to cast MenuItem to Listbox?

I got the same exception as you did. After looking closer at the code, this feels like a situation where you would get

"The event 'Click' cannot be specified on a Target tag in a Style. Use an EventSetter instead."

I'm not sure why that doesn't apply here.
Anyway, using an EventSetter works

<Setter Property="ContextMenu">
    <Setter.Value>
        <ContextMenu>
            <MenuItem Name="AddProjectElementMenuItem" Header="Add">
                <MenuItem.Style>
                    <Style TargetType="MenuItem">
                        <EventSetter Event="Click" Handler="AddProjectElementMenuItem_Click"/>
                    </Style>
                </MenuItem.Style>
            </MenuItem>
        </ContextMenu>
    </Setter.Value>
</Setter>

I had to face this weird situation myself. There is an easy way to get over it, you have to clean and rebuild the project and the exception will go away.

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Hope this helps.