Just to add to this (and off the bat I will say I’m pretty much drawing on Francine Hirsch’s Empire of Nations): late Tsarist-era/early-Soviet ethnographers tended to be influenced by then-fashionable German concepts of Voelkisch-ness (which saw ethnic groups in a semi-scientific light), while communist thinkers themselves were more influenced by French concepts of nationality (ie, that nations were the result of a collective conscious identity).
As noted above, Soviet anthropologists did engage in physical anthropology, notably study of blood types. Nevertheless, debates in the 1920s over how to define ethnic groups largely rejected using any sort of physiological traits or blood origin, but rather focused on group consciousness (although, for example, in the 1926 Census, if a respondent didn’t have a nationality to self-identify with, the census takers could prompt them with “objective” determinants like kinship, region, religion or language).
Soviet anthropology and ethnography in the 1920s and 1930s, adapted fairly easily to the new regime, and found general Marxist concepts of stages of development amenable to how it presented different ethnic groups: ie, if culture was superstructure to an economic base, and if a people’s economy was determined by environmental factors, then the perceived “primitiveness” of, say, Siberian pastoralists was explainable in terms of environment and economy, rather than race.
Soviet anthropology was heavily influenced by Edward Burnett Tylor and his stages of development, notably his concept of “survivals” (which were basically customs or habits from earlier stages of development that were carried through to later stages of development by force of habit). These “survivals” (perezhitki in Russian) became a particular target for explaining public health problems that, in the West, would be explained by eugenics at the time - so for example, veiling in Central Asia was seen as a cultural practice that promoted the spread of syphilis and the weakening of health of children.
The rejection of “race science” by Soviet anthropology became explicit after 1930, and in no small part for international political reasons: it was championed by Nazi Germany, and so the USSR, for political and ideological reasons, put a lot of effort into disproving biological determinism, hierarchies of race, and the idea of racial “degeneracy”. It should be noted that this actually wasn’t a debate from afar: the Soviets and Germans conducted joint anthropological expeditions in the USSR from 1920 to 1937, so for instance there were expeditions to the Chuvash and to the Soviet Far East where the German anthropologists examined physical characteristics and the Soviets examined social and cultural characteristics based on economic conditions. The Soviets also attempted to work out theories of “non-racist race science”, largely based on the work of Franz Boas, that saw group physical characteristics as caused by isolation, and migration/contact with other groups as a means of undoing these characteristics.
Of course, it’s worth noting that sometimes these efforts went to the other extreme. There was interest in proving the ideas of Lamarck, namely in the concept of “acquired characteristics” - ie that people could change their physical characteristics and pass on those new characteristics to their offspring. This fit very well into the concept of Homo sovieticus, and the ability of the Stalinist revolution to change the culture and very physicality of Soviet people. Of course, the absolute extreme of this would be Trofim Lysenko and his charlatin “vernalism”. Lysenko specifically worked with botany (not anthropology), but he was favored by Stalin for developing these Lamarckian views, and because of his political favor and his persecution of rivals, he essentially destroyed Soviet biology for a generation.
A final note: outside of academia, there were, especially in Central Asia and Siberia, some vague ideas of race, but these tended to be expressed in terms of “European” and “Asian” or “Mongolic”, and of course where Europe ends and Asia begins in Russia is a whole separate topic in itself. For what its worth, these were largely more cultural than racial concepts, so in Central Asia, there was very much a distinction between local Asians and “Europeans”, but “European” included Russians, other Slavs, Ashkenazi Jews, and Tatars.
Overall, the Soviet Union was far more interested in developing and cataloging nationality (and the stages of development those nationalities were in) than race.