In fact, here's an explicit example with $\chi(M)\not=0$: Let $M$ be the 1-point compactification of the hyperelliptic Riemann surface given in the affine plane by $$ y^2 = x^5-1. $$ This is a smooth Reimann surface of genus $g=2$ and hence $\chi(M) = -2$. The holomorphic differential $$ \omega = \frac{dx}{y} = \frac{dx}{\sqrt{x^5-1}} $$ has only one zero (over the point $p$ where $x$ and $y$ have poles of order $2$ and $5$ respectively). Consequently, the metric $$ g = \omega\circ\overline{\omega} $$ is flat and is a smooth $(0,2)$ tensor that vanishes only at $p$.
Similar hyperelliptic examples can be constructed for any genus greater than $0$.