Is this a logical fallacy? If this is a logical fallacy, what kind of logical fallacy is this?

Yes, this is a logical fallacy known as the "genetic fallacy." The genetic fallacy occurs when a claim or argument is judged based solely on its origin or history rather than its current meaning or context. In this case, the argument claims that because mochi was created before the country was officially called Japan, it cannot be considered a Japanese dish. This reasoning is flawed because it ignores the cultural and historical context that has led to mochi being recognized as a traditional Japanese dish today. The name or status of a country in the past does not determine the present cultural identity or associations of its food.

Aside from the Genetic Fallacy mentioned in the other response, the argument remains weak because there was no effort to define "Japanese" in the context of a dish.  If "a Japanese dish" is defined as something like "one that comes from a place called Japan at the time", then there would be a case to be made for Mochi not being Japanese ... however, that's a pretty unusual definition for "Japanese dish".

Without a common understanding of the terms used, it's difficult to expect any level of agreement.