Let me give a proof using the deformation theory of holomorphic maps developed by Horikawa [Journal Math. Soc. Japan 25]. This can be seen as a purely analytic proof in the spirit of Kodaira's "deformations of complex structures".
Set $X:=\Sigma_{g_1}$ and $Y:=\Sigma_{g_2}$, and fix a non constant holomorphic mapping $f \colon X \to Y$.
Now
Let us denote by $\textrm{Mor}(X, Y)$ the space of holomorphic maps from $X$ to $Y$. It is an analytic space, whose tangent space at the point $[f]$ coincides with the space of first-order deformations of $f$ is given by f$, namely $H^0(X, f^*T_Y)$.
Since $Y$ is of genus $g \geq 2$, we have $\deg T_Y <0$, so $H^0(X, f^*T_Y)=0$.
This means that the morphism $f$ is rigid, in other words there are only finitely many first-order deformations of $f$ up to composition with automorphisms of $X$.
But it is well known that $|\textrm{Aut}(X)| \leq 84(g(X)-1)$, so there are only finitely many first - order deformations of $f$ at all.
This shows that every component of the space $\textrm{Mor}(X, Y)$ is a point. In general, it can happen that $\textrm{Mor}(X,Y)$ has countably many components; in this case, however, it has only finitely many of them, since the possible degrees of $f$ are bounded from above by the Riemann-Hurwitz formula. This implies that there are finitely many choices for $f$.
If you do not like deformation theory, there exists actually a purely analytic (and completely different) proof of a definitely strong result, the so called Kobayashi - Ochiai Theorem:
Theorem. Let $X$ be a Moishezon space and $Y$ a compact complex spece of general type. Then the set of meromorphic maps from $X$ to $Y$ is finite.
From the proof, that is a combination of techniques and uses in an essential way the Schwarz lemma, I refer you to the original Kobayashi-Ochiai paper [Meromorphic mappings onto compact complex spaces of general type, Inventiones Math. 31 (1975)]
