Rachael,
Based on the engine pics you posted (great job even if you are blonde) the tube or cup you refer to on your breather is connected to the heat shield you show. I will try to explain.
If you notice the "door" inside your breather has springs. It is opened and closed by the heat that runs off your engine and up through the heat shield, into the tube (that connects to your "cup") opening or closing the door in your breather. It all works with the choke and carb too (although I can't explain how!) I believe the heat shield also trys to keep your starter from getting too hot and not being able to turn your engine. Most mustangs have taken all that off and run a different breather, unless they are in competition or shows. Finally, Your part you posted last (during your blonde moment) looks modified to me, because it looks incomplete, unless it is like Doug says and is from a later mustang. In that case, it probably is modified. It has nothing to do with california emissions, that is a whole other topic and california did have extra emission equipment on their engines. Anyway, I hope this helps and it is not a critical piece for your car to run.