Here is a counter-example when $A$ is regular local of dimension $4$ and $n=3$, the first non-trivial case. Let $A = k[[x,y,z,t]]$, and $P$ be a prime ideal of height $3$, say $P=(x,y,z)$. Let $M=\Omega P$, the syzygy of $P$. Obviously, $M$ is also $\Omega^2 (A/P)$.

The depth of $M$ at various localizations can be computed easily by tracking depth along short exact sequences. In particular, $M$ is $(S_2)$ since any second syzygy module over $A$ would be. Since $depth(A/P)=1$, it follows that $depth(M)=1+2=3$. On the other hand, when you localize at $P$, $M_P$ is a second syzygy of the residue field in a $3$-dimensional regular local ring, so $depth(M_P)=2$.

Here is a counter-example when $A$ is regular local of dimension $4$ and $n=3$, the first non-trivial case. Let $A = k[[x,y,z,t]]$, and $P$ be a prime ideal of height $3$, say $P=(x,y,z)$. Let $M=\Omega P$, the syzygy of $P$. Obviously, $M$ is also $\Omega^2 (A/P)$.

The depth of $M$ at various localizations can be computed easily by tracking depth along short exact sequences. In particular, $M$ is $(S_2)$ since any second syzygy module over $A$ would be. Since $depth(A/P)=1$, it follows that $depth(M)=1+2=3$. On the other hand, when you localize at $P$, $M_P$ is a second syzygy of the residue field in a $3$-dimensional regular local ring, so $depth(M_P)=2$.

PS: as can be seen in the comments below the question, there are different definitions of $(S_n)$ in the literature, see What is Serre's condition (S_n) for sheaves? However, the notion I assumed for this answer is the most restrictive one ((1) in the one cited), so it applies to others as well.