Does "back off" mean "go away from" here?

No. Back off is an idiomatic command to leave something alone or halt what they are doing (in this case: an attempt to stop the EU regulations regarding the trademark protection which wouldn't allow US manufacturers of said food stuff from legally using those terms in question).

BTW, failed to understand yours "which wouldn't allow US manufacturers of said food stuff from legally using those terms in question". The question is the use of the word "from" there. But I surely get "allow legally using..."

I also failed to understand "only for products made there".
Does it mean "except for products made in the United States"?

oristarA wrote:

I also failed to understand "only for products made there".
Does it mean "except for products made in the United States"?

Only for products made in Europe. In the European Union we have legal protection of "designation of origin", so that product names such as Champagne, Chianti, Gorgonzola, etc have to be from those regions and produced to a certain standard.

EU and external countries' similar laws can be mutually enforced under bilateral trade agreements, including US regulated designations (Vidalia onions, Florida orange juice, Idaho potatoes and Tennessee whiskey are examples).