He was a "Hebrew", just like Avraham Avinu was. The Torah calls Avraham "Avraham the Hebrew". His children were part of the Hebrew nation, aka Israel. Not only that, even before the Torah was given at Mt. Sinai, there already were some commandments that were kept. For example, all men from the 12 tribes were circumcised. They knew they were different, they knew they were one nation. They knew they believed in the G-d of Avraham, Isaac and Yacov. They were part of one distinct nation, and at various times in history that nation was called nation of Hebews, Israel and then Jewish.
I am not sure when "Jewish" started to be used though.
But I would say based on this, we know Moshe was Jewish. Of course, I am not too well versed, I am sure others will provide a better explanation.