Specifically in the USSR areas of Ukraine/Russia.
Soviet identity papers and other bureaucratic forms often did require Jewish ethnicity to be labeled on them, but this was true for other ethnic groups within the USSR. The introduction of internal passports in 1932 had a an infamous line five in which called identified the passport holders' natsional'nost' (nationality). Likewise, birth certificates had some type of self-registration on the part of the parents. Although the state was ambivalent about accepting "Jewish" as a discreet ethnic category throughout the lifespan of the USSR, Jewish remained a recognized category available in the Soviet family of nations.
In theory, the Soviet state was building a society in which ethnic categories would be subsumed by class ones, and thus being Jewish or Ukrainian would be irrelevant to the New Soviet Man (the common term used at the time). Categorizations like Line 5 were temporary expedients that served two purposes. Firstly, it allowed the Soviet state to identify the makeup its multiethnic population and help allegedly disadvantaged groups and uplift them. Called korenizatsiya (indigenization, or "Soviet affirmative action" to use Terry Martin's phrase), these policies would encourage native cadres within both the Party and the state and undo the damage one by the Great Russian chauvinism encouraged by the tsars. Korenizatsiya was also part of the rationale behind the erection of the various -stans and oblasts for certain ethnicities in that it gave them a direct territory or "homeland" within the Soviet state. But ethnic categorization also possessed a darker rationale as well. The chaos of the five-year plans as well as the famines created by collectivization created a marked demographic disorder among the Soviet population. This disorder prompted the Soviet state to look much more intently into regimenting and documenting its population as a means of social control. Internal passports controlled movement and allowed the state to identify individuals who were in places they did not belong. While the public rationale behind things like line 5 was to uplift national minorities and protect them, ethnic categories soon emerged as a kind of bureaucratic shorthand to identify allegedly loyal and disloyal elements in the population. Initially, the Soviet police looked to class background to identify these elements, but class was such a malleable category, while ethnicity was not. In particular, the presence of ethnic groups with large populations outside of the USSR, such as Poles, were particularly suspect in the 1930s.
This Janus-faced approach to ethnicity put Soviet Jews in an anomalous position. On one hand, the Soviet state eliminated many of the onerous tsarist regulations on Jewish occupations and movement and allowed Jews into positions of power in the government. The state's official stance on antisemitism was that this form of racism was a tool of monopoly capitalism to divide the working classes, and the USSR's anti-antisemitism became even more prominent with the rise of Hitler. A good many Soviet Jews in the interwar period did embrace a form of Soviet Jewishness that was more secular in orientation and squared Judaism with Marxism. But at the same time, the state's line on antisemitism did not eliminate antisemitism within the USSR and there remained strong strands of antisemitic sentiments among society and government. In terms of korenizatsiya, the Jewish section for the Communist Party was the only nationality section that was explicitly temporary. A Jewish background also became a liability during the Great Purges because it was much harder to erase pre-1917 connections with Jewish socialist groups (the Bund) or Zionists. In contrast, their gentile colleagues often found it easier to fudge their pre-1917 past to fit the mold of someone favorable to the Bolsheviks. Additionally, there still remained widespread antisemitic tropes associating Jews with rootless cosmopolitanism and other vices antithetical to the Soviet state, which made ascribed Jewish identity a liability during the Purges. Mania for uncovering spies and saboteurs often dovetailed with antisemitism that identified Jews as others and aliens. To add insult to injury, the presence of many Jews in the upper echelons of the NKVD, who themselves were purged as the Terror wound down, meant Soviet Jews also got blamed for the Purges even when they numbered among its victims.
But as terrible as the Great Purges were, it was only in the postwar period that the Soviet state began to hew to a more openly antisemitic line. The defeat of Hitler ensured that anti-antisemitism was no longer a major factor in Soviet ideology and the war saw the loosening of punishments for antisemitic outburst. The Great Patriotic War's sanctification of Great Russian nationalism cemented trends that began in the mid-1930s that placed Great Russians as the elder brother in the Soviet family of nations. Although korenizatsiya did not wholly go away, there was much less effort put in by the state to uplift its nationalities. For Soviet Jews, the paranoia of the late Stalin years increasingly turned antisemitic with purges of Jews from top positions of the Soviet military and government. Although the USSR recognized Israel, Stalin did not allow emigration to the Jewish state and recognition was coupled with a degree of official domestic repression of Jews. All explicitly-Jewish state organizations were closed and the state anti-cosmopolitan campaigns played with various antisemitic dogwhistles.
Despite this new round of postwar repression, the antisemitism of the Soviet Union never reached that of the Third Reich and the Soviet government was never openly antisemitic, preferring codewords like cosmopolitan or Zionist. This meant that the Soviet Jewish community that survived the war, and a good many did not, was able to go about their lives. The Thaw saw an easing up on the state's antisemitism and there was a good deal of acculturation that occurred within the Jewish community. But the damage of Stalin's postwar campaigns had been done, and there was a glass ceiling for Jews within the Soviet state. Acculturated Jews could ascend some parts of the Soviet system, but seldom break through to become a boss or other figure with weight. This lack of clout was important in the Khrushchev and Brezhnev economies where clout allowed for access to scarce consumer goods and other perks in the chronically underperforming command economy. The state's anti-Zionist rhetoric of the Cold War added further strains to the relationship of Soviet Jews to the state as it became increasingly apparent that Jews were a begrudgingly tolerated alien presence in the USSR. Emigration to Israel and to various diaspora communities, which began under the Thaw, became one of the realities of Soviet Jewish life in the 1960s and 70s. The Soviet state was ambivalent about emigration, but allowed in it in the name of normalizing relations with the West.
So the postwar experience of Soviet Jews was quite mixed. German genocide had eliminated the core of those Soviet Jews who had been very heavily Sovietized and Soviet Jews found themselves something of strangers in their own lands. After the dark period of the immediate postwar, Soviet Jews could live their lives, but with little hopes for a brighter future. Discrimination continued in overt and covert forms, but typically did not cross the line into open repression.