I think a lot of people would disagree that there is such a thing as a Jewish race, to start with. This gets at the question of what exactly is a race, as well. I think it most likely has something to do with your genes and family lineage, without getting too deep into it.
Unsure of this myself, I have googled and found that some writers with prominent search ranks say that Jews were always considered to be a race until after the holocaust. I still don't think it's remotely correct to say "synonymous" even here.
Except it isn't. There's a clear distinction between Jews as an ethnic group and Judaism the religion. Hence frequent criticism of and among various Jewish movements without it being anti-Semitic.
Not really. You're considered a Jew by other Jews if at least one of your parents is Jewish (your mother in most sects, your father in a few other sects). That's very tribalistic. So, what's the common communal identifier of Jews? Jewish culture, and interwoven with that, Judaism. You can't separate the religious from the cultural component of that communal identity, and you can't separate Jewish culture from Jewish people.
This also shows in how Israel regulates its immigration: both people with ethnically Jewish roots (at least one of your grandparents) and people of Jewish belief where ethnic roots hadn't been established at that time (biggest example: Ethiopian Jews) are accepted and granted citizenship.
Conversion of people to Judaism are relatively rare, and are really a more recent phenomenon. Jews don't proselytize and often discourage conversion. Samaritans, the "sister" religion of Judaism (both derive from the Israelites from before the destruction of the first temple), are similar but even more extreme in that. How more tribalistic than that can you get?
However, Jews also have disproportionately high levels of atheism, secularism and other forms of irreligion, which means much of the Jewish culture (Judaism) is relegated to irrelevance, but the ethnic identification nonetheless remains.
Jewish culture as such is certainly interwoven with Judaism, but becoming increasingly syncretic. Look at the large number of hilonim in Israel.
