Was the USSR a nation or more like the European Union?

Neither. The Soviet Union was nominally a federation of republics and other administrative units like the Autonomous Soviet Socialist Republic (ASSRs) that voluntarily joined into a political union, but there was a strong centralizing imperative within the USSR's government. Soviet propaganda stressed the voluntary nature of the union and pushed issues like Belarus and Ukraine joining the UN as independent states. In the 1920s, the authorities in Moscow delimited the administrative boundaries of the new state based on the national principle, hence the proliferation of territories based upon ethnic groups. The glue that held this federation together was to be the Communist Party (CPSU). There was a program called korenizatsiia, roughly translated as indiginization (Terry Martin called it Soviet affirmative action) that sought to promote native cadres of the CPSU within these ASSRs and SSRs. The leaders of these cadres had some degree of power within their turf, but because such power came through their connection to the CPSU, there was a limit to their autonomy. Moscow still controlled foreign policy, the basic legal system, and, very important in a planned economy, the distribution and allocation of resources. Now also bear in mind that the relative autonomy of the SSR and ASSR leadership varied somewhat throughout the existence of the USSR; there was quite little leeway under Stalin, but more lax conditions in the Thaw and Brezhnev periods.

Edit: One other clarification here, being a "Soviet citizen" in the USSR was a supranational identity, at least in theory. Soviet passports provide evidence of this as they would have a box for nationality along with residence. Although Soviet nationality policies often had an ad hoc character, the Soviet authorities consistently operated under the assumption that an individual's identification with a national or ethnic group would become secondary to their identification with the Soviet project. In practice, things never quite worked out in the way the Soviet leadership planned. Nationality was a slippery construct and in cases such as Central Asia there was a strong impulse of the Soviet state to incubate national identity among the various and diverse ethnic groups. The borders of the SSRs were often far from apparent and local SSR leaders would try to expand their territory on the basis of the national principle. Finally, the USSR became increasingly Russocentric from the 1930s onwards; partly out of pragmatism in which Russian and Cyrillic were a lingua franca for the diverse territories, but also out of a subtle form of Russian chauvinism wherein Russia became the elder brother of the family of Soviet nations. However, one should avoid the temptation of looking at the Soviet Union as the Russian Empire 2.0, principles like korenizatsiia never really went away in the USSR and dividing up the USSR on the basis of ethnicity created an alternative form of identity other than the supranational Soviet forms which increasingly had a Russian tone about them.

Source

Smith, Jeremy. Red Nations: The Nationalities Experience in and After the USSR. Cambridge: Cambridge University Press, 2013.