I don't see that the Text is set in VS2005, it did in VS2003 (and I think it was annoying). It is the IDE that is naming the controls, it does this based on the class name of the control and adds a counter to ensure no control gets the same variable name, that have to refer to the control instance in forms generated code.
The place to do Designer related work with the control if not in the class template would be through an ControlDesigner for the class, but if you try to do this for example in the Initialize override you will still only have the type name (it also don't change generated code), the IDE has not yet changed the name of the control at this point, actually while the property displays this value is does not take effect until runtime when the control is instantiated. In code view you also use this name, but here it only refer to the variable that in runtime will point to the instance, and again Intellisense will play along and is a feature of the IDE. So I don't know if it is possible for the control to find this name at design time. You also see in the properties pane it is written "(Name)", the paranthesis probably mean it is 'special'. If you use the OnMouseEnter+Leave overrides to set the Text you'll also see the control Name remains same as control class name anytime at design time.