No its not a fake product; its real
Outside of the US, the carry cases were manufactured under license. Sometimes the license holder was the company who also owned the right to make the ponies but often in countries which were stocked by Hasbro themselves they were under license to another toy company. So in Greece the carry cases are made by El Greco under license from Hasbro, just like the ponies, while in Denmark they were under license to K.E. Mathiasen because there was no local MLP license.
However what was common was that rather than set up to produce the item themselves, the licensee would simply buy in the product from someone who was set up to manufacture. In Europe the main company set up for manufacture was DAG in Italy. So those who held the right to sell and distribute the carry cases in their own countries would simply "buy" the item from DAG - hence the made in Italy stamp. DAG produce like for like with just the name of the carry case changed to the appropriate language. I know there are also Dutch and French versions.