The traditional legal definition of a Jew is well-known: the child of a Jewish mother or a convert. Sure, there is a little fuzziness around the edges as Orthodox Jews in Israel in particular debate what makes a kosher conversion, and whether conversions can be retroactively revoked. But both Orthodox and Conservative Jewish institutions share their commitment to this legal definition. Jewish identity is for all intents and purposes black and white, verified or falsified with objective markers.
Yet as both the U.S. census and Susan Fendrick in a recent article remind us, life is not lived in black and white with objective markers. These legal definitions matter greatly to bean counters and lawyers (or rabbis or others in the role of lawyers), and to people only when they run up against bean counters and lawyers. As Fendrick sensitively suggests, it is not that legal definitions of Jewishness are wrong, only that they don't adequately reflect lived experience. As for the U.S. census, they can't figure out what to do with "Race" when many people, flummoxed by the check-off boxes, liberally select multiple identities.
Jews in antiquity had, if anything, even more fluid identities as Jews. Prior to the Rabbis, there were few necessary objective markers of Jewishness (circumcision for men was one, but even it was not sufficient). For most of the people most of the time this would never have been a problem; they had few if any encounters regarding their Jewishness with bean counters or lawyers. There were, of course, some exceptions: when the Romans leveled a tax on all Jews throughout the Empire following the disasterous revolt that led to the destruction of the Temple in 70 CE, Jews had to be counted. We have little idea how they did this (although Martin Goodman has a provocative articleon its ramifications). Jewish communities locally handled their own "conversions", and it is intriguing to consider how converts, their families, and different Jewish communities might have had very different understandings of who they actually "were".
In this vein it is interesting to consider the famous CCAR's statement on patrilineal descent, which requires at least one Jewish parent and an affirmative act of identity. This is legally a nightmare: it would allow for cases in which Jewish identity was stripped and in which the children of two Jewish parents would not be considered Jewish. I doubt that there have ever been more than a few cases of either of these scenarios, if only because there is a tacit acknowledgement that although this is meant as a legal definition it is a poor one. It is, in fact, more descriptive than prescriptive, a comment on the way that Jewish identity is actually enacted today, with or without legal definitions, in living color rather than black and white.