In your question you show the spatial part of a general isotropic and homogeneous metric.
Even though in such a space there exists spherical symmetry, you are free to express the same metric in, e.g., Cartesian coordinates. The spatial part of a flat such metric then reads
dx^2 + dy^2 + dz^2.
This might look more familiar to you and no direction (none of $x$, $y$, or $z$) are special wrt the other. If no direction is special we have isotropy.
Expressing the Cartesian coordinates in terms of spherical coordinates
x &= r\cos(\phi)\sin(\theta\\
y &= r\sin(\phi)\sin(\theta)\\
x &= r\cos(\theta)
and computing the differentials, leads to the form you quote in your question (with $f(r)=r$; rescaling $r$ or replacing it by a real $f(r)$ can lead to non-flat spaces).
That you find the coefficient of $d\phi^2$ to carry $\theta$ dependence simply is a result of your coordinate transformation.