It happens from time to time. I expect if you got a database of games at master level or above to search for identical end positions it would find a couple of dozen perhaps - just guessing. At lower levels, there will be more because novices are more likely to fall into opening traps set by their opponent.
However, it seems you're looking for some statistical answer based on first principles. I'm not sure that's very meaningful in the context of practical play because players tend to follow well-trodden opening paths to various depths.
I often hear that no two chess games are the same. This isn't completely accurate because I could, for instance, perform a Scholar's Mate or Fool's Mate on my opponent and then my friend could perform the same moves exactally like I did with his own game. I think you can say that given a greater number of moves in a chess game, the less likely it is for that game to be replicated exactally move for move because of all the branches of moves you produce. My real question is this: how probable is it for two chess games to be identical in every way after 10 moves, and how do you calculate that probability? I personally believe that through the years, even with a game 25 moves or more, there had to have been at least one game that replicated it move for move. I know I can't prove this scientifically; it's just a belief. What do you think?