This expression is not well-defined. $y$ is a function of two variables. If you want one of them to be $r \theta$, you have to specify what the other one is to know what to keep constant.
As a simpler example, suppose $y = \theta$. If I want my two variables to be $u = r \theta$ and $v = r$, then $y = \frac{u}{v}$ and $\frac{\partial y}{\partial u} = \frac{1}{v} = \frac{1}{r}.$
However, if I want my two variables to be $u = r \theta$ and $v = \theta$, then $y = v$ and $\frac{\partial y}{\partial u} = 0.$
Partial derivative notation is ambiguous. In addition to specifying the variable that changes, it ought to also specify the variables that don't change, but for whatever reason it doesn't.