Your condition implies that there is a nondecreasing function, $\phi_a\colon\mathbb R_{\ge0}\to \mathbb R_{\ge0}$ such that $$|a-x|_S=\phi_a(|a-x|_M).$$

One can reformulate it the following way, if you fix a point $a\in S$ then the angle between chord $[ax]_M$, $x\in S$ and the tangent space $T_xS$ depends only on the distance $|a-x|_M$.

This is quite strong global condition.

In particular if $S$ is a hypersurface then all its ponts are umbilical. In higher codimension, at each point, the absolute value of the curvature vector in all directions has to be the same.

Your condition implies that there is a nondecreasing function, $\phi_a\colon\mathbb R_{\ge0}\to \mathbb R_{\ge0}$ such that $$|a-x|_S=\phi_a(|a-x|_M).$$

One can reformulate it the following way, if you fix a point $a\in S$ then the angle between chord $[ax]_M$, $x\in S$ and the tangent space $T_xS$ depends only on the distance $|a-x|_M$.

This is quite strong global condition.

In particular if $S$ is a hypersurface then any pont is umbilical in the strongest sense ― all its principle curvatures are equal. In the higher codimensions, at each point, the absolute value of the normal curvature vector in all directions has to be the same.