I've never heard of that law applying to restaurants. But if you buy a ready meal in a supermarket and it's, let's say, "italian sausage pasta" - you can be 100% certain that the sausage in it is from Italy, guaranteed.
Here in the US, "Italian X" or "French Y" just means that's the style. There's plenty of space on most labels to indicate a country of origin, and that's exactly what they do: foods made in Italy clearly state on them "Product of Italy" or similar. In fact, some food being made in Italy or France or anywhere in Europe is generally a plus over here, so they're not shy about putting that on the label.
In short, we don't need laws limiting what you can call things, with regard to country of origin, as long as product labels clearly indicate the country of origin. I know full well the $1.00 Chobani "Greek yogurt" isn't actually made in Greece, but some Greek-named expensive yogurt at the fancy grocery store I can guess probably is, and then after checking the label a bit more closely and seeing "Product of Greece" or "Made in Greece", I know it is.
Now, if some lotion says "Aloe Vera Lotion", I expect it to be made with real aloe. I really shouldn't have to check the ingredients list (if it even has one; it's not a food so I'm not sure that's required).
