Is "Jewish" a race? Only something that can be genetically inherited?
No. There are ethnic groups that have been categorized as races which were thought to be almost entirely Jewish and there are people of "Jewish descent" (defined by nazis as a set of parameters regarding how many grandparents practiced) who do not practice.
In the US it became prevalent to ask people what their mother's maiden name was as a way to identify if they were Jewish but that information is rather esoteric and over time the question has become a security option. In nazi Germany, a lot of effort was put into deciding who was and who was not Jewish, and I don't even feel like getting into that.
In Spain, the inquisition forced a lot of Jews to convert to Christendom but very little if any attention was ever paid to descent. It was simply forbidden to practice a religion that rejected Jesus Christ. However, those families who feigned conversion but secretly practiced, maintained records of lineage and were known as Marrano. This was a slur but became an accepted identification that we claimed in defiance of those who would use it on us.
The Ashkenazim are an ethnic group with many distinctive genetic markers who have been found to have inhabited Lithuania for centuries but there are many overlaps with Greek genetic markers. The point is, it is very obvious that people have converted to Judaism for centuries and then maintained records of bloodlines. It did not begin as a race. One does not have to be born a Jew to be a Jew.