Short answer:
“Black privilege” definitely existed in the USSR. Up until the late 1980s when the crumbling of Soviet rule threw open our borders to a huge variety of international travelers, a black person in the streets of Moscow almost always belonged to a tiny privileged minority.
Long answer
Below, a front of a Beryózka grocery shop in Moscow somewhere in the mid-1970s. Those in lawful possession of Western money could buy at such places most of what was available at the time in Europe or the US. Beryozkas were for foreigners and those of Moscovites who had savings from their overseas visits and stays as diplomats, sailors, trade reps, spies, and suchlike.
Invalyúta (“hard currency”)! The salt of the earth, the key to all riches, the joy of their lives! The rest of us were relegated to the sad, crowded, chronically understocked places for lowly ruble commoners.
Together with foreign tourists, the total market of Beryozka shops consisted of some tens of thousands of lucky guys with their families and friends. But an imported car like the 1963 Ford Galaxy in the photo was something else altogether! A very, very rare bird.
The car carries a plate that says “D” (diplomatic staff) and “061” (Republic of Sudan).
Moscovites interested in Western cars whispered to each other the names of a few Soviet Olympians (like Secretary-General Brezhnev or the gravel-voiced actor and singer Vladimir Vysotsky) who held such imported beasts in their garage. Even for the nomenklatura elite, this was out of this world. For mere mortals, the wettest dreams stopped somewhere at a spanking new GAZ-24–95, the vehicle you see at the back of the photo.
But for the blacks we used to see around in Moscow, driving such cars seemed to be as natural as for us a trip in the metro.
Below, black students in the USSR learn how to throw snowballs. The Western clothes they wear would turn any male Moscovite Blenda-white with envy. I was a teenager in the 1970s. A getup like this would make me an instant star in the dating scene. (Don’t even get me started about their stylish tan and pearly teeth. Mine sucked big time.)