In the plane, solving the Laplace equation with Dirichlet conditions is to find a function $u:\overline\Omega\subset\Bbb R^2\to\Bbb R$ such that: $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in\Omega \\ u(x,y) = d(x,y), &(x,y)\in\partial\Omega \end{cases}$$
Verbally, we want a harmonic function $u$ defined in the domain $\Omega$, whose values coincide with those of some $\mathcal C^0 $ function $d$ on the border $\partial\Omega$. An easy version of the problem is when $\Omega = B_1(0)$ (the unit ball). Indeed, we are able to procure an analytic (in the sense that it's given by an explicit formula) solution, whose expression isn't important here, we just care that it exists.
Now say we wish to examine existence of solutions to the Laplace equation, where $\Omega$ is any ($\mathcal C^0$) simply connected region in $\Bbb R^2$ (except $\Bbb R^2$). You can probably see where this is going. Let $$\varphi:\Omega\to B_1(0)$$
be holomorphic (in the sense that the component functions $f,g$ make $f+ig$ holomorphic) with holomorhpic inverse (idem). Also, I'll require that $\varphi$ extend to $\partial\Omega$ continuously, and such that $\varphi(\partial\Omega) = \partial B_1(0)$. This is assured by a stronger version of the Riemann mapping theorem. Now consider the Dirichlet problem $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in B_1(0) \\ u(x,y) = (d\circ \varphi^{-1})(x,y), &(x,y)\in\partial B_1(0) \end{cases}$$
This is exactly of the type that I first mention, so it has a solution $v$. I now claim that $v\circ \varphi$ is a solution to the Dirichlet problem $$\begin{cases}\partial_{xx}u +\partial_{yy}u = 0, &(x,y)\in \Omega \\ u(x,y) = d(x,y), &(x,y)\in\partial\Omega \end{cases}$$
The boundary condition is immediate, since on $\partial B_1(0)$, $v=d\circ \varphi^{-1}$. Therefore on $\partial\Omega$, $v\circ\varphi = d\circ\varphi^{-1}\circ\varphi = d$.
For the PDE, we use that since $v$ is harmonic (i.e. it satisfies the Laplace equation), there exists a harmonic conjugate $w$ such that $v+iw$ is holomorphic. Therefore $(v+iw)\circ\varphi$ (now interpreting $\varphi$ as complex) is holomorphic, and therefore its real part $v\circ\varphi$ is harmonic*, i.e. it satisfies $(\partial_{xx}+\partial_{yy})(v\circ\varphi) = 0$.
By the Riemann mapping theorem, we were able to solve the Laplace equation with Dirichlet conditions over a much wider range of domains, using only that we know a solution in the unit ball. Of course there are much more potent ways of approaching this particular equation, that give even more general results, but I think it's a nice example of an application regardless.
*It's easy to check that if $f+ig$ is holomorphic, then $f,g$ are harmonic, by using the Cauchy-Riemann equations
Best Answer
The fundamental group provides some necessary conditions. As you say in the OP, if I take one region with a hole in it and one without, they have different complex analytic properties. This will clearly prevent a bihilomorphism. One simple reason is because a biholomorphism is necessarily a homeomorphism. In fact, even a diffeomorphism, because holomorphic functions are not just complex differentiable but analytic. Another reason is that biholomorphic maps should preserve the existence of primitives, basically by the chain rule.
Also as you note, the Riemann mapping theorem says that in the case where $\Omega_1$ and $\Omega_2$ are simply connected, there are no further complex analytic invariants - they are all biholomorphic to one another. On the other hand, the story is more complicated for non-simply connected domains.
There is a version of the Riemann mapping theorem for regions with holes in them. These are sometimes called 'multiply connected' regions I think. It is a topological theorem that the only possible fundamental groups of connected planar regions with regular boundaries are free products of $\mathbb{Z}$, because the only such regions are homotopy equivalent to wedges of circles. Ahlfors proves in his Complex Analysis book the following theorem:
Given a multiply connected region $\Omega \subset \mathbb{C}$, there is a biholomorphism $F$ mapping $\Omega$ to a multiple slit region, which means an annulus with $n-2$ angular arcs removed each at some radial distance from the center and of some angular length.
The radii of the overall annulus, and the length of the slits are determined by studying the Dirichlet problem on $\Omega$, but can in some cases be determined explicitly. There is a lot we don't know about finding explicit biholomorphisms between regions, so it can be difficult to tell in general if two regions are biholomorphically equivalent. But the upgraded version of the theorem would state that two multiply connected regions are biholomorphically equivalent if they have the same invariants, in the sense that they both map to the same multiple slit region. There is a lot we don't know about finding explicit maps between regions, so I think I will stop here.