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.
Other clarifications:
In using spherical coordinates $x^\mu = (x^0,x^1,x^2,x^3)=(t,r,\theta,\phi)$, $\phi$ and $\theta$ are coordinates, not indices. In (consistent) index notation, where $x^3=\phi$ and $x^2=\theta$, you want to compute $$\Gamma_{33}^2 = 1/2 g^{2\mu}\left(\frac{\partial}{\partial x^3} g_{\mu 3}+\frac{\partial}{\partial x^3} g_{3 \mu} - \frac{\partial}{\partial x^\mu} g_{33}\right);$$ here $3$ and $\mu$ are indices; moreover on the previous expression there is a sum over $\mu$ implied.
Since $x^3 = \phi$, then $$\frac{\partial}{\partial x^3}=\frac{\partial}{\partial\phi}$$ and so on. They are just different names. So: $$\Gamma_{33}^2 = 1/2 g^{2\mu}\left(\frac{\partial}{\partial \phi} g_{\mu 3}+\frac{\partial}{\partial \phi} g_{3 \mu} - \frac{\partial}{\partial x^\mu} g_{33}\right).$$
Now it comes the slight abuse of notation. Since it is easier to think and remember the 'real' coordinates rather than confusing indices, we write $\Gamma_{\phi\phi}^\theta$ rather than $\Gamma_{33}^2$, $g_{\mu \theta}$ rather than $g_{\mu 2}$ and so on; I would then write $$\Gamma_{\phi\phi}^\theta= \Gamma_{33}^2 = 1/2 g^{\theta\mu}\left(\frac{\partial}{\partial \phi} g_{\mu \phi}+\frac{\partial}{\partial \phi} g_{\phi \mu} - \frac{\partial}{\partial x^\mu} g_{\phi\phi}\right).$$ Remember that you have still an implied sum over $\mu$. Probably in the particular metric you are studying there are some symmetries that allow you to restrict to just one term, the one with $\mu = 2$ (but this could also not happen in other cases), then $$\Gamma_{\phi\phi}^\theta=1/2 g^{\theta\theta}\left(\frac{\partial}{\partial \phi} g_{\theta \phi}+\frac{\partial}{\partial \phi} g_{\phi \theta} - \frac{\partial}{\partial \theta} g_{\phi\phi}\right),$$ with no sum implied anymore.
So now that you have clarified the context I can complete the answer by saying that in your question there is a mistake:
You did not put $\mu = \phi$ for $\mu=3$, but rather $x^\mu = \phi$ for $\mu =3$, or $x^3=\phi$.