As in so many other ways, Judaism is unique. A person who possesses an interest in "converting" to Judaism quickly confronts the question of what she or he is doing. Is she simply embracing a belief structure, as above? Is she embracing an identity transcending a mere belief structure - somehow becoming one of the Jewish people? What exactly is it that happens when she immerses herself in the waters of the mikveh in the presence of a bet din, says the ritual blessings, and is deemed to have completed the final step of a halachically valid conversion? From the fact that her children will be deemed Jewish, we can infer that more has occurred than a mere acknowledgment of shared theological beliefs. But, when she emerges from the mikveh, does she, a "Jew-by-choice", truly share an identity with a Jew-by-birth? Or, are they two distinct identities, both somehow Jewish? Or, is only the latter identity Jewish at all?
I realized keenly that the conversion question leads naturally to the more fundamental question, "What is Jewishness?" when I had a long discussion with an Israeli friend who does not believe that true conversion to Judaism (assumption of a Jewish identity) is possible for someone born a Gentile, Talmudic enthusiasm for the concept notwithstanding. I do respect this view. She was trying to explain to me that no matter how knowledgeable a Gentile becomes about the traditions, customs, and beliefs of Judaism - how fluent in Hebrew - how observant in practice - how assimilated into a community of observant Jews, there would still be a gap that could never be bridged, which she classified as the gap between Jew and non-Jew.
The problem, for proponents of her view, is to explain what makes a Jewish person (i.e. one born to a Jewish mother) who has no relevant Jewish religious or cultural upbringing, "Jewish" in some way that the earnest would-be convert is not. And to provide the tautological response - that the non-observant/ignorant Jew is Jewish purely by virtue of his matrilineal descent - is to beg the question of what, exactly, that purely genetic link provides.
If one puts the question, "What is Jewishness?", to people who identify as Jews-by-birth, they will answer it in an infinite number of ways (thus confirming the age-old, "Two Jews, three opinions."

The question is further complicated by the presence of many individuals who identify as Jewish but do not satisfy the traditional matrilineal test. (For example, people with a Jewish father/non-Jewish mother, or people with a Jewish father/mother-who-converted-but-not-according-to-halacha.) What makes them Jewish? If self-identification, is this in some way different from a convert who, having completed the conversion process, self-identifies as Jewish? From a convert who invokes Shavuot 39a, suggesting that the souls of converts were present at Sinai?
Perhaps, then, Jewishness is the state of having a Jewish soul. Regardless of the extent of a born Jew's Jewish background/knowledge level, he or she would still have a Jewish soul. Perhaps then, upon emerging from the mikveh, a convert is granted a Jewish soul (as some traditions suggest). If so, it is hard to imagine a more weighty proposition.
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On a personal note:
I find this question uniquely frustrating. For the past thirteen years, I have verbalized a desire to "be" Jewish, or have felt a certainty that at some point in the future I will be Jewish. This...dissonance, for lack of a better word...arose years before I knew anything about Jewish religious belief (aside from the texts shared with Christianity), persisted and intensified for the 5-6 year period in which I studied for conversion under rabbinic supervision, and continues, frustratingly, despite the complete incompatibility of my current agnosticism with Jewish religious teaching. For this reason, metaphysical notions suggesting that the souls of gerim were present at Sinai, or the Kabbalistic notion that a "spark" of a Jewish soul exists in many Gentiles, some subset of whom actually convert, almost come as a relief to me. These explanations, which I view as transcending rationality, offer to provide some explanation for what otherwise appears to be an irrational form of dissonance.
But this personal frustration is neither here nor there - it's just keeping me from bed at 2:40 am. I am interested in people's answers to the general question posed in the title of this thread, whatever form they might take.