Well, Judaism is certainly a religion (obviously). But there are many Jews who consider themselves Jewish in more of an ethnic way than in any religious way. I would agree that there may be multiple groups; Ashkenazic Jews certainly have a shared ethnic background, even though there are differences even within that group. Sephardic Jews share an ethnic background, as you say. And then other groups like Persian and Yemenite Jews are sort of their own ethnicities, as well as groups like Ethiopian Jews.
So, perhaps you could say Ashkenazic Jewish is an ethnicity. This just points to why these designations are somewhat arbitrary. Because, as I said before, many people in the US consider themselves Jewish and have a Jewish identity but do not practice Judaism as a religion. So what do they identify with? The ethnic background (or at least the food 😛). But, as you point out, there is not ONE Jewish ethnicity. I would also suggest that there isn't ONE hispanic ethnicity, either. OTOH, my friends who are Sephardic Jews share a great deal of ethnic background with me, and I'm an Ashkenazic Jew. However, this might be due simply to the religious aspects, and it is very difficult to seperate the two.
indo, I'm not sure what your mother's being Jewish has to do with ethnicity. That's a purely religious concept: if your mother is Jewish, you are also Jewish according to Jewish law. Of course you could also convert to Judaism (again, a religious concept), and in that case you would be Jewish without your mother being Jewish. However, within the realm of Americans who identify as ethnically Jewish, there are certainly those who do not have a Jewish mother. From the perspective of Jewish religious law they are not actually Jewish (a religious identification), but they consider themselves Jewish (as an ethnic identification).