I think that there is some problem with your notation.
Disclaimer: what follows are conventions, so they could be not universally respected, although I think this is what is mostly followed.
Reading $$ \frac{\partial g_{\mu\nu}}{\partial x^\mu} = \partial_\mu g_{\mu \nu} $$ in its literal sense, it is not a relativistic expression, since you are contracting two lower indices. So in this case no sum is implied: $\mu$ is fixed. Now you can do two different things.
If you set $\mu =\phi$ you are just relabelling the index and so you can write $$ \frac{\partial g_{\mu\nu}}{\partial x^\mu} = \frac{\partial g_{\phi\nu}}{\partial x^\phi} = \partial_\phi g_{\phi \nu} .$$
Not being a relativistic expression you can specialise in some frame of reference where you can choose particular coordinates, like spherical coordinates, in which you call $x^\mu = \phi$ for some fixed index $\mu$, for instance $\mu =2$. In this case $\phi$ is not an index but is is a coordinate. Then you'd have $$ \frac{\partial g_{\phi\nu}}{\partial \phi} = \partial_\phi g_{\phi \nu} = \frac{\partial g_{2\nu}}{\partial x^2} = \partial_2 g_{2 \nu} .$$
On the other hand, if by your writing you mean $$ \frac{\partial g_{\mu\nu}}{\partial x_\mu} = \partial^\mu g_{\mu\nu}=\sum_{\mu=0}^3 \frac{\partial g_{\mu\nu}}{\partial x_\mu}= \sum_\mu \partial^\mu g_{\mu\nu} $$ (note that the position of indices, different from yours!) this is a relativistic expression and there is a sum implied unless differently stated. Then you can rename the index $\mu$ calling it $\phi$ because it is a dummy index, and you get $$ \frac{\partial g_{\mu\nu}}{\partial x_\mu} = \frac{\partial g_{\phi\nu}}{\partial x_\phi} =\sum_{\phi=0}^3 \frac{\partial g_{\phi\nu}}{\partial x_\phi} $$ In this context an expression like $$ \frac{\partial g_{\mu\nu}}{\partial \mu} = \frac{\partial g_{\phi\nu}}{\partial \phi} =\sum_{\phi=0}^3 \frac{\partial g_{\phi\nu}}{\partial \phi} $$ is not conventionally defined.