So the key idea here, is what population, and what are the players like
If we assume that the player is a random bot with 1,000 games, and we want to find the point where it is more than 50% likely that the bot hasn’t played this move, then even if we count on rotational symmetry, then the first move is 55 for the first move, 360 for the second, and 359 for the third.
so the first move happens with an occurrence of 1/55, so the probability that this random bot has played this move is almost 100% in 1,000 games
the first and second move occurrence is 1/19800, then the probability that the bot has played both of these moves (in that order) in 1,000 games is 49.2%, so we’ve probably found a unique game
But I suppose some people want some level of significance, so we’ll go until 5% or less
So the third move is 1/7108200, and so the probability that the bot has played it (in that order) in 1,000 games is 0.01%, so you almost definitely have a unique game after three moves.
In order to make any further inquiries, we need to have a specific data about the population in question, which usually implies having either a) a complete database of all the games in question or b) some assumptions about how people actually play.