In this Wikipedia article on Winding Numbers, under the sub-section "Complex Analysis," is described the following:
In complex analysis, the winding number of a closed curve C in the complex plane can be expressed in terms of the complex coordinate $z=x+iy$. Specifically, if we write $z=re^{i\theta}$, then $$dz=e^{i\theta}dr+ire^{i\theta}d\theta $$ and therefore $$\frac{dz}{z}=\frac{dr}{r}+id\theta=d[\ln r]+id\theta,$$
and the sub-section goes on a little before ending. My question is - why have they used the product rule in the differentiation of $z=re^{i\theta}$, when $r$ is not specified as a function of $\theta$, for this is how the first equation must have been reached ?
As $z = re^{i\theta}$, $z$ is a function of the variables $r$ and $\theta$, so the rule being used could be described as a multivariable chain rule (for differential $1$-forms), rather than a product rule:
$$dz = \frac{\partial z}{\partial r}dr + \frac{\partial z}{\partial \theta}d\theta = e^{i\theta}dr + ire^{i\theta}d\theta.$$
This is just a multivariable extension of the rule $dy = \dfrac{dy}{dx}dx$ that is often seen when learning about integration by substitution.