Because of the rich combinatorial possibilities of the game, it may be extremely unusual to have games repeated move for move, but I don't think it is impossible, any more than it is "impossible" for anyone to win the lottery more than once. (Who is the lucky four time lottery winner?)
I believe that I have seen articles where games matched games played decades earlier though I can't think of any off hand. (Soltis maybe?)
I think that you are also wrong in thinking that nobody cares enough to create a huge easily searchable database of non-master games. Chess.com's Opening Explorer was an attempt to do exactly that. Each player was supposed to be able to look through the database to find patterns in their own play. The fact that they haven't been able to make it work for a couple of years reflects the difficulties in handling that unwieldy mass of data from the hundreds of millions of games played here to date. (I think they are still trying and they make announcements that it will be up and running every once in a while but who knows?)
If they ever did get it working, it would probably not be set up to search for identical games but I imagine that you could start by searching for identical ending positions and work backwards.
It has been brought up before in other threads, and there have been some identical Master games(draws reached at relatively early positions, probably pre-planned). And, of course, there are many people who have lost to Fool's mates down through the ages...
But I got into a discussion with a friend where he insisted that it would be impossible for two chess games to be entirely identical barring some of these events. I thought that this was patently absurd...
Let's say that we cannot find exactly identical Master games that go 20+ moves... But we are talking about a database of perhaps 5-10 million, and we are also talking about the highest level of play.
What about chess.com, where correspondence chess games we now have 100,000,000+ correspondence chess games and an even larger amount of live games... Now think of all the sites.
Now just think of the number of games starting with the Sicilian... And especially consider that, with tons of non-Masters playing, there's going to be people making the same silly misakes and having some sloppy & amateurish game development. That, of course, includes me, all day long.
I think it is inevitable, just because of the sheer numbers, tat there will not just be identical games... But that there is a constant stream of identical games that end because of poor positions / poor play resulting in checkmates & resignations before we hit move 35-40+ and the openness of the board, the remaining number of pieces, etc. really start to totally differentiate and individualize games.
My friend's position was just that the number / diversity of moves was too great...
Basically, the ideal proof would be being able to produce 2 chess games, played with 4 different and unrelated players with no connection, that had made the exactly identical moves and were above move 20. I think the only issue is that
(a) I don't think people care enough to create easy access, easily searchable millions & millions of game databases of non-master games.
(b) I woldn't know how to search for identical games affectively.
Thoughts?