To add on, how is it taught in modern day Russia and other countries, such as Japan, Poland and Italy?

Modified from an earlier answer of mine

Although antisemitism was not unheard of in the Soviet Union, it played only a contributing factor at best in the postwar downplaying of the Third Reich's crimes against Jews. Discussion of the Holocaust in the USSR was not less forbidden so much as it was downplayed and minimized.

The Soviet approach to war commemoration throughout the postwar era emphasized that Soviet Communism was both the primary victim and target of fascism. Thus massacre sites like Babi Yar, a ravine in which upwards of a 100000 Jews were shot, were not unknown in the Soviet Union, but were instead commemorated as areas in which the Germans massacred thousands of Soviet civilians. This is of course functionally true, the Jewish victims of the Einsatzgruppen were indeed citizens of the USSR, but this generalization enfolded the Nazi’s Jewish massacres into a wider narrative of Soviet victimization. The initial draft of the discovery of Babi Yar for the Soviet press indicated that a majority of its victims was Jewish, but the official memorandum reporting the massacre t deleted these specifics.

The rationale for this minimization goes beyond just the Soviet desire to be the primary victim of Nazism. The Soviet ideological conceptualization of antisemitism was that it was a superstructure of the wider antipathies towards Marxism within the capitalist West. In Soviet orthodoxy, antisemitism was a tool used by the enemies of Marxism, along with religion, racism, or nationalism, to prevent a support for Marxist-Leninist movements. The Nazi apparatus for extermination of Jews was something this Soviet understanding of antisemitism was unprepared for. The Soviet state was also very ambivalent towards Judaism in general. Soviet Jewish policy gravitated between two poles that revolved around the central question “Is Judaism primarily an ethnicity or a religious identity?” If it were the former, then Jews were an ethnic group that deserved protection within the Soviet brotherhood of nations. However, Stalin’s policy increasingly veered towards the latter definition and self-identifying as a Jew was out of place within the Soviet Union as it was a cultural/religious identity. Subsequent Soviet leaders also followed this Stalinist tack, especially as the Soviets became more publicly opposed towards Zionism during the Cold War to curry favor in the Middle East. Consequently, asserting that Jews were a victim independent of the wider actions of the Third Reich stood against the grain of the Soviet stance towards Judaism.

This is not to say that some cracks in this state-centric memory did not appear during the postwar period. One Soviet poet, Yevgeny Yevtushchenko wrote “Babi Yar” in 1961 that connected both the Jewish nature of the massacre with the wider policies of extermination such as the death of Anne Frank. Yevtushchenko got into a bit of hot water for this poem, with Khrushchev critiquing the poet for political immaturity. Other cultural depictions of the war recognized Jewish suffering, but were much more low-key in such acknowledgements. For example, the 1965 film Ordinary Fascism has a harrowing scene in Auschwitz, but does not specifically identify its victims as Jews and asserts that had the Germans been successful, Himmler would have expanded the camps into the Soviet Union. But it would take Glasnost and the collapse of the Soviet Union to undermine this edifice that prioritized Soviet suffering over all others.

There was much the same trend throughout the Eastern bloc. In general, the Eastern bloc commemorative strategies did tend to conflate Judeocide with a wider program of fascist barbarity to suit purposes of national martyrdom. Naturally, national martyrdom was a tad more complicated in the USSR in which official discourse did have to play lip service to the brotherhood of nations within the Soviet government. As early as 1944, the Soviet Extraordinary State Commission to Examine and Investigate German-Fascist Crimes had received explicit instructions not to emphasize Jewish identity of the victims and suppress any evidence of Ukrainian collaboration in the commission of these crimes. The first commemorative sites at Minsk, which prior to the German invasion was very much a Jewish city, contained references to Jewish murder, but the specificity of this commemoration of the Jewish ghetto and murder sites shifted in the postwar period to generalizing the victims as Soviet citizens. Ilya Ehrenburg and Vasily Grossman collected material for a book detailing an extensive list of German actions against Jews and even though the USSR printed the book in 1946, all copies were immediately sent to a warehouse where they remained unread and were eventually destroyed in a warehouse fire. In this environment, the Soviet's treatment of Masha Bruskia was typical and exemplified the erasure process. Even though her very Jewish name was known, the widely-circulated Soviet reproductions of her hanging simply called her "the unknown partisan."

A similar process occurred in Poland. The initial postwar meetings to preserve Auschwitz and open up a museum there planned for an extensive section on the biological racism and antisemitism of the Third Reich, but the higher ups in the Polish Communist Party vetoed this and instead tried to create Auschwitz into a symbol of Polish repression. The Polish parliament framed its Auschwitz commemoration as preserving a place where "Poles and citizens of other nationalities fought and died a martyr's death." The state lavished attention on Auschwitz-I in the first postwar decades, where most Polish prisoners were located, but comparatively little upon Auschwitz-II, the site of mass extermination. The focus of the Polish government on Auschwitz-I as well as some of the initially sloppy preservation efforts has provided no end of grist for Holocaust denialists who contend that Auschwitz is a site ideologically contaminated by communist propaganda. But contrary to the assertions of Holocaust denialists, the most historically suspect of the Auschwitz State Museum's efforts more or less downplayed the death of Jews at the site.

This ambivalence about the Jewish identity of Hitler's victims often extended to textbooks and other pedagogical tools in the Eastern bloc. A Czechoslovakian textbook in the 1950s referred to Auschwitz and Theresienstadt as holding thousands of "democrats," to be murdered. The 1953 official Hungarian textbook for secondary school talked of the double repression of both the Germans and Horthyite fascist and made frequent reference to "martyrs of the international working class movement," but never once mentioned the word "Jew" and relegated all those that did not fit into its political definition of martyrdom as generalized "victims of fascism." Similarly, popular histories in the GDR often emphasized the heroic nature of communist resistance to fascist crimes, often to the detriment of even acknowledging other groups' suffering. Buchenwald and Sachsenhausen became important state pilgrimage sites for the state youth group, Freie Deutsche Jugend, but these sites' emphasized KPD suffering and heroism rather than guilt and contemplation. The Sachsenhausen obelisk features the red triangle and its attending statue features a pieta of Soviet soldiers helping cloth the emaciated KPD prisoners. The SED poster announcing the commemoration featured the flags of Hitler's victims, including Israel, but they are dwarfed and subordinate to the red triangle. '

This process of blurring and organized forgetting continued through the Thaw. Increased contact with the West and Western Jewish organizations caused a slight shift in commemorative practices in the Eastern bloc. The Treblinka Memorial included Jewish iconography such as a menorah, but official pronouncements of the commemoration ceremony described it as a site remembering the "800,000 citizens of European nations" murdered there. In the Soviet Union, the Thaw allowed for more overt acknowledgement of Jewish suffering, but within limits. Shostakovich's 13th Symphony initially had lines from Yevtuschenko's Babi Yar poem in its libretto , but the censors replaced explicit lines "I feel myself a Jew," to the more innocuous "Here Russians lie, and Ukrainians/Together with Jews in the same ground." The 1979 edition for Babi Yar in the Great Soviet Encyclopedia described the massacres as :

At the end of September 1941 the German fascist occupiers shot 50,000–70,000 persons—mainly Jews—in Babi Yar; for the next two years the so-called Syrets death camp operated there. Communists, komsomol members, members of the underground, prisoners of war, and others were imprisoned there. In August-September 1943 the fascists, retreating from Kiev and trying to cover the traces of their crimes, destroyed the camp and exhumed and burned hundreds of thousands of corpses in ovens; the ashes were scattered in the vicinity of Babi Yar. At the end of September 1943 there was a revolt of 330 condemned prisoners who were working at the ovens; 15 survived. In October 1976 a granite obelisk was erected at the site of the mass executions.