For a multivariable function $y=f(x,z)$, is $\frac{dy}{dz}$ a well-defined object?

Question

Almost forgot basic calculus which were taken more than 10 years ago and feel confused about the following point: For a multivariaable function $y=f(x,z)$, we know that $\frac{\partial y}{\partial z}=\frac{\partial f(x,z)}{\partial z}$ is the partial derivative of $y$ with respect to $z$, and it captures the partial effect of $z$ on $y$ when $x$ is fixed. I'm wondering in general is $\frac{dy}{dz}$ a well-defined object if $y$ is the dependent variable of a multivariable function? If yes, what does it equal to? How to interpret it? I seem to have some vague impression that $\frac{dy}{dz}$ represent the effect of $z$ on $y$ when $x$ is not fixed, but I'm not sure. If you could help clarify this with intuitive explanation, that would be great!