Interpretation of $S$-ideal class group

Here are a few reasons why one shouldn't expect $Cl_S(k)$ to be related to $Gal(k_S / k)$, or even to $Gal(k_S^{\mathrm{ab}} / k)$ where $k_S^{\mathrm{ab}}$ is the maximal abelian extension unramified outside $S$.

Firstly, the Galois group of $k_S^{\mathrm{ab}} / k$ will generally be quite large; e.g. for $k = \mathbf{Q}$ the extension $k_S^{\mathrm{ab}} / k$ is infinite as soon as $S$ is non-empty. On the other hand $Cl_S(k)$ is always finite.

Moreover, the field $k_S^{\mathrm{ab}}$ gets bigger and bigger as you enlarge $S$, while the S-ideal class group gets smaller -- for $S$ sufficiently large it will be trivial.

If you are familiar with the isomorphism of global class field theory between $Gal(k^{\mathrm{ab}} / k)$ and the idele class group $\mathbf{A}_k / \overline{k^\times k_\infty^\circ}$, you can easily see what's going on. The group $Gal(k^{\mathrm{ab}}_S / k)$ corresponds to the quotient of the idele class group by the image of the subgroup of $\mathbf{A}_k^\times$ given by $$ \left( \prod_{v \notin S}\mathcal{O}_{k, v}^\times\right) \times \left(\prod_{v \in S} 1 \right).$$ The group $Cl_S(k)$ corresponds to the quotient by $$ \left( \prod_{v \notin S}\mathcal{O}_{k, v}^\times\right) \times \left(\prod_{v \in S} K_v^\times \right).$$ So these are very different beasts, unless $S$ is empty.

Relationship between Cl_S(k) and the splitting of primes : Cl_S (k) is the quotient of Cl(k) by the subgroup generated by the classes of the prime ideals in S. If we identify by class field theory Cl(k) with the Galois group over k of the maximal abelian unramified extension of k (the so called Hilbert class field), then the aforementioned subgroup is isomorphic to the common decomposition subgroup of all the primes in S.