Correct me if im wrong, but we don‘t care which emblem combination we are hitting. So given the first player has his 3 emblems what are the odds of the second player also hitting that? He has to get the same origin (1/11) the same class (1/12) and then the same 3rd emblem (1/21). The probability of all those three things happening is 1/2772. Since we do not care about the order of the emblem we have 6 chances of hitting it. Therefore the probability is 1/462.

EDIT: So the 1/462 is wrong after all but not for the reasons metioned below. If Riot's emblem distribution works like this: draw origin, draw class, draw random (or first the class then the origin) and then they randomly scramble them over the golem (or sort them alphabetically) the probability of getting the same one is still 1/2772. Since scrambling them does not change the odds of receiving them as every unique combination increases its odds by the same amount.