"Race" is a cultural construction.
If we really tried to pin down what race was from a biological perspective, it would be... very challenging, to say the least. What we conventionally mean by "race" is people who look similar, where skin colour plays an unusually large role. In the USA, if you had a black grandparent and four white grandparents, you probably look more stereotpyically white than black, but you'll be called black. For no other reason than cultural US norms, really. That's to illustrate how kind of stupid and arbitrary it can be. And within any one race, like "white people", there is staggering diversity - even in skin colour, really. Lily white Scandinavians, olive Mediterranean, etc.
So when we consider that we could define race by other means, other genes, etc. (there are 30,000 genes, after all, and only a very small proportion of them govern skin colour), why not "Jewish"? We could here take the similar idea of "bloodline", which is not so tied to looks. And we would expect very considerable diversity of Jews, just like there is in any other "race". Although in practice most Jews are still (by genetic analyses) quite similar to the other populations of the Levant, which is a testament to the fact that they bred with each other so much rather than the Europeans or whoever else they lived amongst all those centuries. Although to be fair, a large reason for that interbreeding was probably that they were massively discriminated against and exogenous marriage thus deeply unfavoured by Jews or gentile. That said, I suspect a lot of Jews did marry exogenously, but also converted, their bloodlines subsumed into the general milieu and culture/religion abandoned.