First of all, apply the AM-GM inequality as follows:
$$\left(\frac{a}{b} + \frac{c}{d}\right) + \left(\frac{b}{c} + \frac{d}{a}\right) \ge 2\sqrt{\frac{ac}{bd}} + 2\sqrt{\frac{bd}{ac}} = \frac{2(ac + bd)}{\sqrt{abcd}}$$
Continuing to apply the AM-GM inequality, then
$$\left(\frac{a}{b} + \frac{c}{d}\right) + \left(\frac{b}{c} + \frac{d}{a}\right) \ge \frac{2(a + c)(b + d)}{\sqrt{abcd}} \ge 2 \cdot \frac{2\sqrt{ac} \cdot 2\sqrt{bd}}{\sqrt{abcd}} = 8$$
The above inequalities turn into equalities when $a = c$ and $b = d$. Then the condition $(a+c)(b+d) = ac+bd$ can be rewritten as $4ab = a^2 + b^2$. This is equivalent to $a/b = 2 \pm \sqrt{3}$.
Hence, $S$ attains value 8, e.g., when $a = c = 1$ and $b = d = 2 + \sqrt{3}$. $\square$