As Zhen mentions in the comments, the local ring of $V(xy)\subseteq\Bbb C^2$ is the localization $\left(\dfrac{\Bbb C[x,y]}{(xy)}\right)_{(x,y)},$ since $(x,y)$ is the maximal ideal corresponding to the point $(0,0).$
An important property of localization is that it commutes with quotients. In this case, we have $\left(\dfrac{\Bbb C[x,y]}{(xy)}\right)_{(x,y)}=\dfrac{\Bbb C[x,y]_{(x,y)}}{(xy)}.$ Since we know that $\Bbb C[x,y]_{(x,y)}$ is the subring of $\Bbb C(x,y)$ consisting of elements $\dfrac{f(x,y)}{g(x,y)}$ satisfying $g(0,0)\neq 0,$ we can see that $\dfrac{\Bbb C[x,y]_{(x,y)}}{(xy)}$ consists of elements $\dfrac{f(x,y)}{g(x,y)}$ where $g(0,0)\neq 0,$ but also where $f(x,y),g(x,y)$ can have no mixed term monomials $x^\alpha y^\beta$ with $\alpha,\beta\neq 0$ with nonzero coefficient.