Is there a standard system of classification of games? Given there are practically infinite number of chess games possible, does there exist a standard system of classifying or naming games simlar to how there are rules for naming a chemical compound uniquely?  There are 10 pieces that can move initially, so I imagined there would be 10 families of games and each game's pgn would be the identity of the game and so on. But the ECO classification system   is quite different than what I imagined. What I am really interested in is this-will a certain family of games would be more interesting than the other? Will certain family of games have less draws or wins with perfect play? Is there more zugzwang positions in a certain family?
Such a system doesn't exist exactly, but a few similar concepts do. There are many reasons, but the main reason is that it's not possible to classify chess games like this in any useful way.
Yes, you could classify them based on the first move -- but who cares? Only 1.c4, 1.d4, 1.e4 and 1.Nf3 are really important, and they often transpose into each other (1.d4 is followed by 2.Nf3 quite often, and 1.Nf3 is followed by 1.d4).
For a system like this to be useful, it needs to have some properties:

It should be based on the games of really strong players, after all they are the ones who care about this sort of thing
Some people have really amazing memories, but even they are quickly overwhelmed by chess. It has to be of a practical size
It has to help you find good moves at the board, as that is really the only thing that matters for chess players

And a whole chess game is quite long. Some parts of it can be categorized, others defy such attempts completely. So in chess it is positions that can sometimes be classified, not entire games.
I can think of three phases that can be treated like this:

The early opening. Theory can be worked out exactly, because the starting position is known. Related lines will have things in common but also important differences, and they mostly just need to be memorized. Opening names like in ECO are mostly so you can find in which book you need to look up your theory.

Most middlegames (not all) can be usefully classified based on the central pawn structure. That's the main thing most players will think of. In e.g. a position where one side has &quot;the isolated queen's pawn&quot; (a pawn on d4 and no pawns on the c- and e- files, after the opening, and the other side say a pawn on e6 and no pawns on the c- and d- files) has a list of standard plans that people know by heart. The actual positions differ, the tactical details will matter, but recognizing lots of types of position like this and knowing the standard plans helps. It is what books like Mauricio Rios' Chess Structures are about.

Very simplified Endgames can again be classified and studied. Many positions with bishop + pawn vs bishop have similar ways for trying to promote the pawn or stopping it, and knowing about them helps (and it helps to decide whether you can safely trade down to this endgame in a particular case), and so on for lots of other material distributions. Many, many books are written about endgames in such a fashion.


But there are also phases of the game in between the early middlegame and the late endgame, and they lend themselves much less to classification. The differences are more important than the things games have in common, here.
And there is no &quot;standard&quot; classification for any of this. It's an art, more knowledge of this sort of thing (and the skill to apply the knowledge) is what separates strong players and even stronger ones.