A Soviet citizen was simply called a "Soviet person" (советский человек , Sovetskiy chelovek) in Russian. However it should be noted that this was not an important phrase in the Soviet Union, and there was the more important distinction of nationality. This was important for a variety of factors, as the rights and privileges of various ethnic groups varied over the 70 years of the USSR.

The number of ethnic groups recognised within the USSR varied, and while I don't have the numbers handy at the moment, there were roughly 120 distinct ethnic groups one could choose for the 1926 census, the first under the post-Revolutionary USSR. This number was condensed in subsequent censuses, but by 1989 there were still around 100 different options one could register as.

One's ethnicity, or nationality to use the Soviet term, was determined by one's parents. Either parent could be used to claim an ethnic identification, but only one could be used, and it was not transferable. This was noted on internal Soviet passports, which either helped or hindered depending on what nationality you were, and where you were living. The era also played a factor, as the 1920s and early 1930s saw a lot more efforts to promote smaller, non-Russian groups (the affirmative action empire, as Terry Martin has termed it), while the Stalinist era saw a reversal and a more Russianised era that ended this. This ignores that come the 1950s the concept of the "New Soviet man," or the more informal "Homo Sovieticus," which was supposed to represent the next stage of socialism, one which moved beyond the bourgeois concepts of nationality. This would be the era when it was promoted, and was even used by Khrushchev in a speech at the 22nd Congress of the Communist Party in 1961 ("In the USSR, a new historical community of people of different nationalities, having common characteristics - the Soviet people. They have a common socialist homeland-the USSR, a common economic base-a socialist economy, a general social class structure, a common world view-Marxism-Leninism, a common goal-the building of communism, many common features in spiritual shape, in psychology.")

Edited to correct translation and add Khrushchev quote