Names like Champagne, Roquefort, Parmigiano-Reggiano and Aceto Balsamico Tradizionale di Modena have a special status under EU law, that these are Protected Designations of Origin, which states standards for the product and gives legal protection regarding use of the name. That limits what you can call "Roquefort" in the EU. These restrictions have not been legally relevant in the US and are still fairly widely ignored especially in the case of Champagne, although Parmigiano-Reggiano and Aceto Balsamico Tradizionale di Modena are not used as generic terms as Champagne or Asiago are. Violations in the EU may involve charges of counterfeiting, false advertising, or fraud. The essence of PDO designation is that it describes a product that is traditionally and wholely made in a geographic region and where the character of the product derives from that region (e.g. the grass that the cows eat). This protection does not operate in the US or Japan (as far as I can tell). There are some specific agreements between the EU and particular countries such as a wine and spirits agreement with South Africa, but PDO really only operates within the EU.
Otherwise, the applicable concept is "trademark". You could not register a trademark for "Champagne" in the US, because that is a generic term, but you could coin a new term like "Gorgonzeddar" for a greenish-orange cheese, and register it. From what I can determine about Japanese trademark law, you similarly could not register "Sake" as the name of a rice-based drink, since it is a general term. If you had an unused name that you want to protect, you can register a trademark, and you would need to register that trademark in every country of interest.