PS maybe I don't understand what you are tracking here. Can you describe exactly what you are looking for? As I understand it, you are comparing the two dice of the first roll of the game, i.e. of the first player to move, with the two dice on the next roll, i.e. the first roll of the second player to move.

Are you counting situations in which the same dice occur, but in a different order, as a match or non-match?

The probability of the same two dice occurring in the same order on the second roll is 1/36. However, the probability of the same two dice occurring in any order is 2/6 x 1/6 = 1/18, as you observe.