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.
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?
Jewish culture as such is certainly interwoven with Judaism, but becoming increasingly syncretic. Look at the large number of hilonim in Israel.