Quasilinear Preference and MRS
The MRS is a function of $x$ and $y$. The fact that $y$ doesn't appear in the formula simply means that this function is constant in $y$. 

The linearity does not imply indifference. It means that an additional value of $y$ is valued equally by the agent irrespective of his baseline consumption. In your example, this contrasts with the marginal utility of consuming $x$ which is decreasing with the baseline consumption since the function $u$ is concave in $x$.