You can't automatically get a "fully qualified" control name that I'm aware of...you'd have to manually build up the name by ascending up the parent tree until you get to the form.
Visual Studio will not allow you to have controls with the same name on the same form...BUT it is actually possible at RUN-TIME to give controls the same name.
If you built a UserControl, though, then you could have the same name controls in each instance of the UserControl and simply place instances of the UserControl into your TabPages.