Looking at older movies (Rocky 4 for example) I get the impression that people (in western media atleast) referred anyone from the USSR as 'russian'. Did the people from USSR find this objectionable? How much did the average USSR citizen identify as 'russian' ? Did this vary from place to place ?
The ones who weren't Russian certainly did. Many forget that the USSR was a political union of fifteen separate countries (I think it was) that attempted to swallow up more than that many ethnic identities. Russia, and its Russians (as well as its non-Russians), was just the biggest of the member states.
In addition to the non-Russian Soviet citizens, as mentioned in other comments, even Russians found it somewhat objectionable because USSR was supposed to be a post-nationalistic (trans-nationalistic?) union, and it was preferable to be called sovietskiy (soviet person) rather than russkiy (Russian).
I even remember reading a book set during the wartime, where the main character was telling off an Englishman for it. She was saying that calling them Russian was appropriate before 1917, and that enough time has passed already that foreigners should get used to calling them Soviets.