When Senator Bernie Sanders thanked supporters for his landslide victory in the New Hampshire Democratic primary, he wistfully reminisced about his upbringing as “the son of a Polish immigrant who came to this country speaking no English and having no money.”

While the crowd cheered, Rabbi Michael Paley of New York was among many Jews watching the speech who were taken aback. He said he was surprised that the Vermont senator had not explicitly described his father as a “Polish Jewish immigrant,” a significant distinction given Poland’s checkered history with its Jewish population.

“Nobody in Poland would have considered Bernie a Pole,” Rabbi Paley said.

Two days later, in a debate with Hillary Clinton, Mr. Sanders referred to the historic candidacy of “somebody with my background” without overtly saying he was Jewish. That prompted the Jewish Telegraphic Agency, a news service feeding the Jewish press worldwide, to ponder, as its headline put it, “People are confused why Bernie Sanders won’t own his Jewishness.”

Mr. Sanders, those who know him say, exemplifies a distinct strain of Jewish identity, a secular offshoot at least 150 years old whose adherents in the shtetls of Eastern Europe and the jostling streets of the Lower East Side were socialists, anarchists, radicals and union organizers focused less on observance than on economic justice and repairing a broken world. Indeed, he seems more comfortable speaking about Pope Francis, whose views on income inequality he admires, than about his own religious beliefs.