$\{ e_i \}$ is an orthonormal basis which has the orthonormal condition as following $$ e^T_i \cdot e_j = \delta_{ij} $$
Not quite. The reason I say this because this statement isn't the most general statement you could make about orthonormal bases for vector spaces (or at least it's not precise enough)
At any rate, the correct general statement is as follows.
Let $\{ e_i\}$ be a discrete (possibly infinite) orthonormal basis for a complex vector space $V$ equipped with an inner product " $\cdot$ ". Then, for any $i,j$ we have that
$$ e_i \cdot e_j = \delta_{ij}. \tag{a} $$
Wait a second. Shouldn't that be $(e_i^*)^T \cdot e_j$? No, and this is because the " $\cdot$" denotes an inner product between two vectors $v, w$---whatever that may be. That is, the dot does not refer to matrix multiplication necessarily, but rather refers to the inner product between two vectors $v,w\in V$.
Note that the case you cited $(e_i)^T \cdot e_j = \delta_{ij}$ is a special case of (1) when we are representing the basis vectors in a matrix representation and (2) when it is a real vector space, since then $e_i^\dagger = (e_i)^T$. It’s precisely when both of these are true that
$$e_i \cdot e_j = e_i^T \ e_j = \delta_{ij}$$
Lastly, Dirac notation is just the notational change
\begin{align} & e_i \to | e_i \rangle\\ & e_i^\dagger \to \langle e_i|\\ & v \cdot w \to \langle v |w \rangle. \tag{b} \end{align}