The only instance I am familiar with is that case during the late 80s when Moscow appointed a man in charge of Kazahstan who was russian and not a kazah which caused mass rioting there and was one of the stepping stones of the collapse of the Union.
But I winder whether this problem was more of an exception or has been USSR finding itself in the same turbulent position about race and ethnicity through it existence as other nations did at its time?
You might be interested in Grigori Aleksandrov's 1936 musical Circus, which attends to American racism in a very interesting away (along with some excellent songs!).
But it's pretty clear the society presented in the film didn't exist. At one point clearly-Jewish actors sing in Yiddish - after the nonexistent Doctors' Plot this scene would be removed, and reinstated only after Stalin's death. Nonetheless American racism became a very prominent political issue in the 1930s, as the 'enlightened' Soviet Union did not suffer the fractures of the West - there are some fantastic propaganda posters of this era.
As far as other forms of racism, indigenous ethnic minorities including the Chechens and Tatars were internally moved betweens the 1920s and 1940s; the WWII era deportation of the former has been recognized as a genocide [1]. Between 1920 (the Polish-Soviet War) and 1939 over 100,000 Poles were executed in Ukraine and Belarus, and many more displaced; this, similarly, has also been argued to be a genocide [2].
Alexander Fadeyev's The Rout (1927) is probably more telling for the racial perceptions of East Asians. The Japanese are vicious (remember the Russo-Japanese War), while the Koreans are at times sycophantic and meek. Koreans were also mass deported in the late 1930s [3], but in the post-Stalin period this was less of a problem; consider, Viktor Tsoi's band Kino, for example.
So, yes, racism was a significant issue in the Soviet Union, but just as your anecdote implies it manifested quite differently than in the West.
[1] https://www.europarl.europa.eu/sides/getDoc.do?type=TA&language=EN&reference=P5-TA-2004-0121
[2] Rogovin, Vadim Zakharovich. Stalin's Terror of 1937-1938: Political Genocide in the USSR. Mehring Books, 2009.
[3] Pohl, Jonathan Otto. "Cultural, Spatial, and Legal Displacement of the Korean Diaspora in the USSR: 1937-1945." The Review of Korean Studies 21, no. 1 (2018): 171-188. [Pohl has several monographs on this topic, but I have not read them, so I unfortunately can't speak to them]