Rock, paper, scissors definitely has an optimal strategy of just choose randomly for each toss. And two players using that strategy are in Nash equilibrium.
What I'm wondering is if that's the same as the game being "solved", or is there a distinction between the two concepts? After all you still can't predict the outcome of the game
Yes, it is solved because it has been proven that no other strategy is superior to random choice on the assumption that you do not rely on the play of your opponent. Say your opponent plays rock the first $20$ times. There is a temptation to assume that your opponent will always play rock, so you should play paper. The game theory I learned considers that out of bounds and passes that question to the psychologists or somebody else. Game theory is predicated on the best strategy for you, either ignoring what your opponent does or assuming your opponent is seeking his/her best advantage. It does not ask that you predict the outcome of one game, but does ask that you predict the average outcome of a long series of games.