Let $S\subset M$ be a (complex) codimension $1$ submanifold. $S\cdot S$ denotes the self-intersection manifold of $S$ in $M$. Then it seems that $\text{Sign}(S\cdot S)$ (the signature of $S\cdot S$) the self-intersection number of the divisor $S$? If so, I wonder why it is true?
I think the manifold $S\cdot S$ is given by $S\cap S'$, where $S'$ is gotten by bringing $S$ in transversal position to $S$ via Transversality Homotopy Theorem.
This is from p.78 of this paper.
You have failed to mention the dimension of $M$. This, of course, matters. If $\dim_{\Bbb C} M = 2$ then your $S \cdot S$ is a finite collection of points (and it makes sense to talk about its intersection number: the signed count of those points.)
The signature of a connected oriented $4k$-manifold $M$ is the signature of the pairing $H^{2k}(M;\Bbb Z) \otimes H^{2k}(M;\Bbb Z) \to H^{4k}(M;\Bbb Z) \cong \Bbb Z$, where the isomorphism uses the orientation of $M$.
When $k = 0$ (so you have an oriented point), there is a canonical isomorphism (no orientations in sight yet) $H^0(M;\Bbb Z) \cong \Bbb Z$, and your pairing $H^0 \otimes H^0 \to H^0$ takes the form $\Bbb Z \otimes \Bbb Z \to \Bbb Z$, with $(m,n) \mapsto mn$. The matrix defining this bilinear form is $\begin{pmatrix} 1 \end{pmatrix}.$
However, we have not followed the orientation isomorphism. The orientation isomorphism $H^0(M;\Bbb Z) \to \Bbb Z$ is identified with $\Bbb Z \to \Bbb Z$, with this map being $+1$ if the point is signed positively and $-1$ if the point is signed negatively. Therefore the bilinear form that signature is, well, the signature of is $$\begin{pmatrix} \pm 1 \end{pmatrix},$$ where the sign indicates whether the point is positively oriented or not.
By definition to get signature of a disconnected oriented manifold you take the sum of the signatures of each connected component. Thus if $M$ is an oriented 0-manifold $\text{Sign}(M) = \# M$, where the last expression means the signed count of points.
In particular you are correct: when $\dim_{\Bbb C} M = 2$, the signature $\text{Sign}(S \cdot S)$ is just the intersection number of $S$ with itself.
In general $$\dim_{\Bbb C} (S \cdot S) = \dim_{\Bbb C} M - 2.$$ When $\dim_{\Bbb C} M > 2$ there is no such thing as the intersection number of $S$ with itself. Instead you get another even-dimensional complex manifold which you can take the signature of.