Pretty much what Agol said, but I wonder if you've seen the generalization of the Schwarz-Ahlfors-Pick theorem (that shows how Schwarz lemma may be understood in terms of the curvature of a suitable conformal metric as you said) to other manifolds of higher dimension. Systems of pseudometrics on complex spaces that satisfy the Schwarz–Pick lemma have been studied extensively. A Schwarz-Pick system is a functor $X\to d_X$ that assigns to each complex Banach manifold $X$ a pseudometric $d_X$
so that the following conditions hold:

(a) The pseudometric assigned to the unit Disk $\mathbb D$ is the Poincare metric
$$d_{\mathbb D}(x_1,x_2)=\tan^{-1}\left(\frac{x_1-x_2}{1-x_1\overline{x_2}}\right)$$

(b) If $X,Y$ are two complex Banach manifolds then
$$d_Y(f(x_1),f(x_2))\le d_X(x_1,x_2)$$ for all $x_1,x_2\in X$ and $f\in \mathcal O(X,Y)$ (holomorphic maps $X\to Y$).

Now $\mathcal O(\mathbb D,X)$ and $\mathcal O(X,\mathbb D)$ provide upper and lower bounds for $d_X$ which correspond to the Kobayashi and Caratheodory pseudometrics. To have a meaningful notion of the Schwarz lemma you would need the Kobayashi pseudometric to be non-degenerate

There is certain notions of hyperbolicity around. Call a complex manifold $X$ Brody hyperbolic if there are no non-constant holomorphic maps $\mathbb C\to X$. Call $X$ Kobayashi hyperbolic if its Kobayashi pseudometric is non-degenerate. At least for compact complex manifolds these two notions are equivalent (sketch of proof here).