You are using Coefficient
with a list as second argument. This usage does not seem to be documented. I don't understand what you are trying to do. However, you can turn your expression into a polynomial with a replacement rule:
expr = 1/(
8 Bi (1 + k)^2 S^2 (Bi + Bi k -
4 k S^2)) (-2 Bi (1 +
k) (-(1 + k) (Bi - 4 S^2) Cosh[2 S (-1 + Y)] +
Cosh[2 S] (Bi + Bi k - 4 k S^2 -
4 S^2 Cos[(Sqrt[Bi] Sqrt[-1 - k] (-1 + Y))/Sqrt[k]] Sec[(
Sqrt[Bi] Sqrt[-1 - k])/Sqrt[k]])) Subscript[A, 2] +
1/(Bi + Bi k - k S^2) 8 Sqrt[
Bi] (Bi + Bi k -
4 k S^2) (Sqrt[Bi] (1 + k) S (Bi + Bi k - k S^2) Y +
Sqrt[-1 - k] Sqrt[k]
S^3 Sec[(Sqrt[Bi] Sqrt[-1 - k])/Sqrt[k]] Sin[(
Sqrt[Bi] Sqrt[-1 - k] Y)/Sqrt[k]] +
Sqrt[Bi] (1 +
k) ((-Bi (1 + k) + k S^2 +
S^2 Cos[(Sqrt[Bi] Sqrt[-1 - k] (-1 + Y))/Sqrt[k]] Sec[(
Sqrt[Bi] Sqrt[-1 - k])/Sqrt[k]]) Sinh[
S] + (1 + k) (Bi - S^2) Sinh[S - S Y])) Subscript[A,
3]);
vars = {Y, Cosh[2 S (-1 + Y)],
Cos[(Sqrt[Bi] Sqrt[-1 - k] (-1 + Y))/Sqrt[k]],
Sin[(Sqrt[Bi] Sqrt[-1 - k] Y)/Sqrt[k]], Sinh[S - S Y]};
Collect[expr /. Thread[vars -> {a, b, c, d, e}], {a, b, c, d, e}]