The answer to this question depends on which time period you are discussing.

It's also easier to address the question by flipping it around: given that the USSR was built out of the remnants of the Russian Empire, where Russian supremacy was imposed and thus assumed to be a matter of fact by many historians, it's easier to understand the USSR by discussing how non-Russian identities were permitted or even encouraged to flourish as a means of reducing the influence of Russians.

The Leninist period

The Leninist vision saw nationalism as a bourgeois construct, designed to artificially separate the proletariat of the world with artificial borders that prevented them from linking up in an international revolution. So, in creating the USSR, Lenin sought to show the world that it was possible for different peoples to preserve their unique national identities while also joining a grand revolution that would unite them in the project of building socialism.

Lenin also feared merely reconstituting a modern Russian Empire, as he feared this would deter other neighbouring countries from embracing an international communist revolution. So he put in place steps to prevent the reappearance of what he called "Great Russian chauvinism" after the revolution.

Lenin's main goal was to devolve power over cultural, social and educational policy to a series of autonomous republics, while Moscow would oversee overall economic development, foreign policy, and defence. Lenin therefore conceived of, and convinced the party to embrace, a constitutional system whereby the local party organisations and state bureaucracies of the republics were nominally independent of Moscow and could set their own policies on questions like educational curriculum, religious and cultural festivals and holidays, official languages, etc.

In the non-Russian regions, Lenin encouraged the promotion of ethnic minorities into positions of power (secretaries of party cells, delegates to the Congress of Soviets, People's Commissars of the governments of the republics). It is true that some regions remained dominated by Russians; for example, even as late as 1930, the Kazakh Communist Party was still 50% Russian... but that was an improvement from its membership in 1917, which was 90% Russian. However, if you looked to Ukraine, by the late 1920s, you would find Ukrainians occupied a narrow majority of positions in the Ukrainian Council of People's Commissars, the Ukrainian Congress of Soviets, and the Ukrainian Communist Party.

Even in the central government, Lenin promoted non-Russians to positions of power. Most notably, he appointed the Polish Felix Dzerzhinsky as head of the secret police, the Jewish Leon Trotsky as Commissar of War, and Georgian Joseph Stalin as Commissar of Nationalities and General Secretary.

Through the Commissariat of Enlightenment, Lenin encouraged the development of social policy that promoted the cultural and national identities of minorities. Party cells and government authorities in non-Russian regions were encouraged to celebrate literature, poetry and folk art forms from national minority groups. Schools were required to teach the basic curriculum in the local language, and were not required to teach Russian. The Commissariat of Enlightenment even helped create written languages for central Asian languages which had no script or written literature - thus helping codify, preserve and spread those languages, and according to historian Robert Service, to outright invent new national identities for certain central Asian peoples.

Census taking and government forms also reinforced national identity - from 1926, all Soviet citizens were required to give an ethnic identity, and census takers and government officials were instructed not to accept generic responses like 'I am a peasant' or 'I am a muslim'. Census takers and Soviet government officials helped group people into neat ethnic categories - encouraging them to self-identify as, say, a Kazakh or Bashkir, rather than a Muslim.

The result was that to ethnic minorities and to the outside world, Lenin could somewhat convincingly claim that the USSR was not a new Russian Empire - it was a federation of peoples united by the shared goal of building Communism. National identity was fostered insofar as it allowed peoples of the USSR to feel their own identity was recognised and treated as unique, but it was never allowed to interfere with the central government's overall economic plans or hopes for international revolution.

The Stalinist period - the return to Great Russian nationalism

The Stalinist period saw a retreat from Lenin's attempts to promote non-Russian identities. Stalin, despite being ethnically Georgian, believed that national identity was fundamentally problematic to the project of 'socialism in one country'. While Lenin's policies were initially preserved in the late 1920s and early 1930s, as Stalin's power grew, the was more of a retreat to the traditional tactics of Russification enforced by the Tsars. For example, by the mid 1930s, teaching of Russian as a second language became compulsory throughout the non-Russian regions and republics of the USSR. Propaganda praised the unifying influence, military achievements and overall leadership of the Russian people. This poster gives a great example of how this focus on Russian greatness was a theme in Soviet propaganda in the late Stalinist period: https://www.alamy.com/glory-to-the-russian-people-a-hero-people-a-creator-people!-1946-artist-image60434682.html

The focus on Great Russian nationalism was especially emphatic during and immediately after the 'Great Patriotic War' against Nazi Germany (1941-45), when Stalin found it easier to play upon Russian patriotism than to appeal to an internationalist vision of resisting fascism and spreading Communism. It also helped that when the Germans invaded in 1941, they quickly occupied all of the western republics like Ukraine. The loss of regions inhabited by tens of millions of people belonging to ethnic minorities reduced the need to address their identities, and thus the 'Great Patriotic War' was fought primarily by ethnic Russians, who were subsequently portrayed as liberators of the national minorities like Ukrainians after the war.

The post-Stalinist period

After Stalin's death, Soviet government policy essentially reached an equilibrium between the Stalinist and Leninist views of the 'national question'. Until the fall of the USSR, it remained clear that Russians were the dominant social, cultural, economic and political force in the union. Thanks to mandatory Russian second language classes in non-Russian regions, the Russian language was firmly entrenched as the lingua franca of the USSR - a pragmatic advantage that helped knit together the union economically, as it enabled greater social mobility (for the Kazakh student who studied in Moscow) and greater economic mobility (for the Russian mechanic who could find a job maintaining tractors on the virgin lands in southern Siberia). However, the Soviet government was also conscious that somewhere around 50% of the union's population were not Russian, and each of the post-Stalinist leaders were also conscious that the USSR sought to project an image to the world of overcoming parochial, inward-looking nationalist attitudes... so the Leninist focus on cultural, educational and social policy that promoted national minorities continued, as did the affirmative-action hiring practices of governments in the republics, and the promotion of non-Russian in the republics' Communist parties.

Main source: Robert Service's A History of Modern Russia. Provides an excellent survey of Soviet history.

Some points also drawn from Orlando Figes' Revolutionary Russia 1891-1991, Stephen Smith's Russia in Revolution 1890-1928 and Sheila Fitzpatrick's The Russian Revolution.

​