On Tarjan's paper "Finding a Maximum Clique"
You definition of $A(C)$ is wrong. It is the set of vertices adjacent to all vertices in $C$:
$$
A(C) = \{ v \in V : \forall u \in C, (u,v) \in E \}.
$$
In particular, if $C = \emptyset$ then $A(C) = V$.
In your example, suppose that $S = \{v\}$. Let's see what $|C| + \|G_{A(C) \setminus S}\|$ amounts two for all choices of $C$:

If $C = \emptyset$, then $A(C) \setminus S = V \setminus \{v\}$, and so $|C| + \|G_{A(C)\setminus S}\|$ is the maximum size of a clique in $G$ that doesn't contain $v$.
If $C = \{v\}$, then $A(C) \setminus S = N(v)$, the set of neighbors of $v$. Therefore $|C| + \|G_{A(C) \setminus S}\|$ is the maximum size of a clique in $G$ that contains $v$.

So in this case, the formula is correct.
When $S = \emptyset$, the formula just states that $\|G\| = \|G\|$.