Let us begin with the complex vector space \begin{equation} V_{z}=\Big\{\omega\in \mathbb{C}[[z,z^{-1}]]dz\ \mid \operatorname{Res}_{z=0} \omega (z)\Big\} \end{equation}
We could define the tensor product of $V_{z_1} \otimes V_{z_2}$.
My question is why $$\omega_{0,2}:= \frac{dz_1\ dz_2}{(z_1 - z_2)^2}$$ does not belong to $V_{z_1} \otimes V_{z_2}$.
Residue of $\omega_{0,2}$ at $z_1 =0 , z_2 =0 $ is zero. What is going here?
The tensor product consists of elements $$f(z_1)g(z_2)\,dz_1\,dz_2$$ where $f$ and $g$ are Laurent series satisfying your residue condition. But $$h(z_1,z_2)=\frac1{(z_1-z_2)^2}$$ cannot be written as a product of a Laurent series in $z_1$ and a Laurent series in $z_2$. Nothing mysterious about that; most functions of two variables are not products of two functions of one variable.