"Is there a way to copy the components from one GUIDE GUI to another and retain the tag names?"
openfig() the .fig file associated with the GUI. copyobj() the appropriate parts of the returned figure into the figure that is going to be the merged result.
Note that the callbacks that GUIDE generates internally all reference the original GUI name, such as
@(src, event) OriginalGUI('openFiles_KeyPressFcn', src, event, guidata(src) )
and those properties are going to remain the same when you copyobj(), leaving you with references to multiple functions. copyobj() will not adjust the saved callback properties for you.