Okay, I don't know what you want then. Let's make this easier to understand:

StructureConstantsList = ArrayReshape[PadRight[Alphabet[], 3^3, Alphabet[]], {3, 3, 3}]

This is

{{{"a", "b", "c"}, {"d", "e", "f"}, {"g", "h", "i"}}, {{"j", "k", "l"}, {"m", "n", "o"}, {"p", "q", "r"}}, {{"s", "t", "u"}, {"v", "w", "x"}, {"y", "z", "a"}}}

You said,

I want to extract ... row2 from each matrix

and I suggested this:

StructureConstantsList[[All, 2]]

(* {{"d", "e", "f"}, {"m", "n", "o"}, {"v", "w", "x"}} *)

That's my best guess for what "row2 from each matrix" means. One of your examples is,

StructureConstantsList[[2, ;;]]
(* {{"j", "k", "l"}, {"m", "n", "o"}, {"p", "q", "r"}} *)

but you said that's not what you want. By the way, that's equivalent to StructureConstantsList[[2]]. Do you want the second row of the second matrix?

StructureConstantsList[[2, 2]]
(* {"m", "n", "o"} *)

If none of these are what you want, then tell me what the correct output is if you use my letter form of StructureConstantsList as input.

Well, I suppose I could write that up, but you don't need to specify the number of sub-matrices in your argument. It will work for any number.

StructureConstantsList = ArrayReshape[PadRight[Alphabet[], 5 3^2, Alphabet[]], {5, 3, 3}];
StructureConstantsList[[All, 2]]
(* {{"d", "e", "f"}, {"m", "n", "o"}, {"v", "w", "x"}, {"e", "f", "g"}, {"n", "o", "p"}} *)

StructureConstantsList = ArrayReshape[PadRight[Alphabet[], 1 3^2, Alphabet[]], {1, 3, 3}];
StructureConstantsList[[All, 2]]
(* {{"d", "e", "f"}} *)

If what you want is a function that takes the lists of matrices as an argument, you can do that easily:

#[[All, 2]] &

Or are you saying that you want the index of the row to be an argument?

RowsByIndex[listOfMatrices_, index_] := listOfMatrices[[All, index]]