This is just a comment, but I don't have enough reputation yet.
It's true for compact hyperbolic curves (=smooth projective curves of genus at least two) for trivial reasons. In fact, if $X\to Y$ is a finite etale morphism of compact hyperbolic curves of degree $d$ and $Y\to X$ is a finite etale morphism of degree $e$ we obtain that $$2g_X -2 = (2g_Y-2)d = (2g_X-2)dd^\prime.$$ This implies that $dd^\prime =1$ and thus $d=d^\prime = 1$.
More generally, the above approach shows the following. Suppose that $f:X\to Y$ is finite etale and $g:Y\to X$ is finite etale, where $X$ and $Y$ are smooth projective varieties. Then, the "Riemann-Hurwitz formula" implies that $$K_X = f^\ast K_Y = f^\ast g^\ast K_X = (gf)^\ast K_X.$$ Maybe this implies that $K_X$ is torsion, but I'm not sure. Of course, I'm assuming $\deg g, \deg f>1$.
It might be more natural to just consider the isomorphism $(gf)^\ast \Omega^1_X \to \Omega^1_X$. You can compute the Euler characteristic of $X$ and $Y$ by taking the $n$-th Chern class ($n=\dim X=\dim Y$), and then you see that the Euler characteristic of $X$ (and $Y$) is zero (if the above surjective map is an isomorphism).