$\|S\|=\sup\{|\langle Sx\;,\;y\rangle |;\;\|x\| \leq 1,\, \|y\| \leq 1\}\,?$

We have first $\|S\|=\sup_{\|x\|\leq 1}\|Sx\|$. Now $\|Sx\|=\sup\{|\left<Sx,y\right>|: \|y\|\leq 1\}$.

Indeed, for $x\ne 0$, then $y:=\dfrac{x}{\|x\|}$ is such that $\|y\|=1$, then $\dfrac{\|Sx\|}{\|x\|}=\left\|S\left(\dfrac{x}{\|x\|}\right)\right\|=\|Sy\|\leq\sup_{\|x\|\leq 1}\|Sx\|$. So $\|S\|\leq\sup_{\|x\|\leq 1}\|Sx\|$.

Now, for $\|y\|\leq 1$, then $|\left<Sx,y\right>|\leq\|Sx\|\|y\|\leq\|Sx\|$, so $M:=\sup\{|\left<Sx,y\right>|: \|y\|\leq 1\}\leq\|Sx\|$.

Now, $\left|\left<Sx,\dfrac{Sx}{\|Sx\|}\right>\right|=\|Sx\|$, and $\left\|\dfrac{Sx}{\|Sx\|}\right\|=1$, of course, if $Sx=0$, this leads to the trivial case.

Additional notes. If $M$ is a subspace of $F$ such that the image, $\text{Im}(S)$ is such that $\text{Im}(S)\subseteq\overline{M}$, in this case, we have \begin{align*} \|S\|=\sup\{|\left<Sx,y\right>|: \|x\|\leq 1, \|y\|\leq 1, y\in M\}. \end{align*} Indeed, as $\dfrac{Sx}{\|Sx\|}\in\text{Im}(S)$, find a sequence $(y_{n})$, $y_{n}\in M$ such that $y_{n}\rightarrow\dfrac{Sx}{\|Sx\|}$. Now, set $z_{n}=\dfrac{y_{n}}{\|y_{n}\|}$. We have $\|y_{n}\|\rightarrow 1$ and hence $z_{n}\rightarrow\dfrac{Sx}{\|Sx\|}$ and that $\left<Sx,z_{n}\right>\rightarrow\left<Sx,\dfrac{Sx}{\|Sx\|}\right>=\|Sx\|$, but $\|z_{n}\|=1$, by the inequality that \begin{align*} |\left<Sx,z_{n}\right>|\leq\sup\{|\left<Sx,y\right>|: \|x\|\leq 1, \|y\|\leq 1, y\in M\}, \end{align*} we deduce that \begin{align*} \|Sx\|\leq\sup\{|\left<Sx,y\right>|: \|x\|\leq 1, \|y\|\leq 1, y\in M\}. \end{align*}

For a fixed $x$, we have $$\sup_{\|y\| \le 1} |\langle Sx, y \rangle| = \|Sx\|$$ by Cauchy-Schwarz (with supremum attained by $y = Sx/\|Sx\|$.

Thus $$\sup_{\substack{\|x\| \le 1 \\ \|y\|\le 1}} |\langle Sx, y \rangle| = \sup_{\|x\| \le 1} \|Sx\| = \|S\|.$$

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If the last equality is not obvious to you, note $$\sup_{\|x\| \le 1} \|Sx\| = \sup_{\|x\| = 1} \|S x\|$$ because if $\|x\| < 1$, then $\|Sx\| < \|S (x / \|x\|)\|$ so the supremum on the left-hand side must be attained by some $x$ with norm $1$. Then, it is easy to show that the right-hand side is equivalent to your definition $\sup_{x \ne 0} \|Sx\| / \|x\|$.

For any $x, y \in X$ with $\|x\|, \|y\| \le 1$ we have

$$|\langle Sx, y\rangle| \le \|Sx\|\|y\| \le \|S\|\|x\|\|y\| \le \|S\|$$

Now, for any $x \in X$ with $Sx \ne 0$ we have:

$$\left|\left\langle Sx, \frac{Sx}{\|Sx\|}\right\rangle\right| = \frac{\|Sx\|^2}{\|Sx\|} = \|Sx\|$$

So taking the supremum over $\|x\| \le 1$ gives:

$$\sup_{\|x\|, \|y\| \le 1}|\langle Sx, y\rangle| \ge \sup_{\|x\| \le 1} \left|\left\langle Sx, \frac{Sx}{\|Sx\|}\right\rangle\right| = \sup_{\|x\| \le 1} \|Sx\| = \|S\|$$

We conclude $\displaystyle \sup_{\|x\|, \|y\| \le 1}|\langle Sx, y\rangle| = \|S\|$.