The USSR was a one-party state, and the state did own and control everything. However, this is fundamentally entirely different from Nazi Germany, which was a combination of corporations and a party owning and controlling everything in a totalitarian Capitalist manner.
It’s reductive to call the USSR fascist, in my opinion. The USSR wasn’t a shining beacon of tolerance and democracy, of course, but at the same time it absolutely was not fascist. The Russian Federation, that rose from the ruins of the USSR, is fascist, and run by extremely wealthy Capitalists.
As for equality in the USSR, gender economic equality was generally high overall, especially when compared to other nations, and especially in the military, but not within government. As for racism, it was apparently much, much better than modern Russia, so make of that what you will, but the state was seemingly mostly composed of white men. Homosexuality was made illegal by Stalin, I do know that.