I'm trying to understand the relation between the adjoint as an operation $T^* $ and the adjoint as a matrix manipulation $T^\dagger$ (transposing and conjugating). So far, I understand that these definitions are equivalent only for linear transformation $T: V \to W$ where we use an orthonormal basis for both $V$ and $W$. However, I'm having trouble convincing myself that this actually works in the case of a nonstandard inner product.
For example, let us define a non-standard inner product by $\langle x, y \rangle = x' A y$. Using this inner product I can surely find some orthonormal basis $\alpha = (v_1, \ldots, v_n)$ of $V$ and $\beta = (w_1, \ldots, w_n)$ of $W$. If I'm not mistaken, however, for the adjoint 'operator matrix' $T^*$ we have $T^* = A^{-1} T^t A$, at least in the real case (see for example this question). Clearly $T^* \neq T^{\dagger}$ even though it seems to me that all requirements are satisfied! What am I missing here?