x:name does not set the name property of a Control.

Now also Code like this:
<Window xmlns="http://schemas.microsoft.com/winfx/2006/xaml/presentation"
        xmlns:x="http://schemas.microsoft.com/winfx/2006/xaml">
  <Canvas>
    <Button x:Name="aa"
            Content="{Binding Name, ElementName=aa}"
            Width="75.2"
            Height="23.2"
            Canvas.Left="85.6"
            Canvas.Top="197.6" />
  </Canvas>
</Window>

works!

src/AddIns/DisplayBindings/WpfDesign/WpfDesign.XamlDom/Project/NameScopeHelper.cs

@@ -38,6 +38,14 @@ namespace ICSharpCode.WpfDesign.XamlDom
 					}
 					if (newName != null) {
 						nameScope.RegisterName(newName, namedObject.Instance);
+						
+						try{
+							var prp = namedObject.ElementType.GetProperty(namedObject.RuntimeNameProperty);
+							if (prp != null)
+								prp.SetValue(namedObject.Instance, newName, null);
+						} catch (Exception x) {
+							Debug.WriteLine(x.Message);
+						}
 					}
 					break;
 				}