How many unique pawn structures are possible?

I've wondered the same thing, and agree with you that there could be some value in classifying board states this way.

Combined with a roster of the remaining pieces on the board (regardless of their positions) the pawn structure specifies a unique board state that, once transitioned out of, can never be reached again in the same game. They form a partial ordering, with no cycles. Every other legal combination of positions for the other pieces can be cycled in and out of any number of times and can always be reached again -- until a piece is captured or a pawn moves (or both.) Those are the only truly irreversible moves in chess.

I think one way to approach determining the number of possible pawn structures is to consider a chess variant I call "pacifist chess" (though it's different from the chess variant of the same name on some chess variant sites I've seen.) No captures are allowed, but otherwise all the rules are the same (including rules about check.) The way the game plays out is that eventually the pawns form a sort of impenetrable wall, or sometimes a wall with one or more "holes" in it. Sometimes, pieces can be trapped inside the pawn wall. Once that happens, the pawn structure of the board cannot change any further. Since pawns can only move forward in pacifist chess, it's easy to generate every possible "earlier" pawn structure that could have led to a given final pawn structure.

Next, consider versions of the game where play proceeds normally for some time, some pieces are captured (and perhaps some pawns moved diagonally one or more columns) and THEN the game switches to pacifist rules. Again, you can (based on the remaining pieces and positions of the now-pacifist pawns) determine the possible final pawn structures and generate the full set of preceding ones. Repeat that for every combination of captures that impact pawn structure (or just shuffle through the roster of other pieces) and you've enumerated every possible pawn structure.

I haven't actually worked all of this out yet, but I think that general approach might be dealing with numbers of a reasonable enough size that computer simulations could work out the rest.

For unique pawn structures I would not recommend going by any formula where you will have 10 to the power of n or any indices calculation. The best way to understand pawn structures is to know multiple openings in. There are more than 2000+ opening variations where the middlegame differs in many aspects along with the piece placement. It is always good to understand the patterns of the pawn structures by playing the openings and learning the middlegame.

This way, you can prepare some of the openings and understand the pawn structures which suit your style. This way, GMs prepare themselves. No one has a direct number of exactly how many pawn structures there are. Not even Kasparov or Carlsen might be interested in that unless you are able to show the perfect differences between each kind of pawn structure.

To classify pawn formations, a good source is chess master books that explore that topic. The book by Boris Persits comes to mind, for example.

I add that it seems impractical as you have been told, to make an exhaustive index of all the combinations that pawn chains can take.

However, it is useful to connect families of openings (classified by ECO, or by open / closed / flank, positional or tactical) and also by the types of endings that can be found according to the pawn structure.

This information has didactic utility for chess plans. There are books that touch on the subject of pawn structures and chess plans.

The pawn chain structures could be described with the characteristics: doubled or solitary / past / advanced / delayed / blocked pawns, V chain or inverted V chain, castling with / without fiancheto and other descriptors that I cannot imagine.

In this way, the number of pawn formations is much less than combinatorial theory can estimate.