So I don't know specifically about the Rubik's cube, but I do know that when the game Tetris was developed, it first spread in essentially an open source format outside of the USSR.

Then once it was extremely obvious that it was a valuable property, various Soviet agencies took control of the copyright, the original creator consent was not required.

The Soviet union did do a substantial amount of trade with the East and so would have been familiar with the process for trade marking, licensing etc.

Hungary while also being a communist country, I suspect would have had a better developed system.