If the set of S-formula in FOL is infinite, then what means a model can interpret this set?

This is addressing the issue as clarified in your comments above rather than the body of the question itself:

The definition of $\models$ is very unlike that of $\vdash$ ... at least, at first glance. In particular, while "$\Phi\vdash\varphi$" is trivially equivalent to "For some finite subset $\Phi_0\subseteq\Phi$, we have $\Phi_0\vdash\varphi$," this is not the case for $\models$. Or rather, this same fact is true but it is not at all trivial: it is the compactness theorem.

Going beyond that, the complete coincision of $\vdash$ and $\models$ - that is, the soundness and completeness theorems - should in my opinion at least be extremely surprising; specifically, we should expect $\models$ to be vastly more complicated than it actually winds up being.

Note that more obvious symmetry between $\vdash$ and $\models$ would make this fact less surprising - so in a sense, the answer to your question "why it is not symmetric between the model-theroy and the sequent calculus" is "because it's more exciting that way!" (This is somewhat tongue-in-cheek, admittedly.)