This is a great question. Someone will come along with a better answer I'm sure, but here's a bit off the top of my head:
1) The Hilbert class field of a number field $K$ is the maximal everywhere unramified abelian extension of $K$. (Here when we say "$K$" we really mean "$\mathbb{Z}_K$", the ring of integers. That's important in the language of etale maps, because any finite separable field extension is etale.)
In the case of a curve over $\mathbb{C}$, the "problem" is that there are infinitely many unramified abelian extensions. Indeed, Galois group of such is the abelianization of the fundamental group, which is free abelian of rank $2g$ ($g$ = genus of the curve). Let me call this group G.
This implies that the covering space of C corresponding to G has infinite degree, so is a non-algebraic Riemann surface. In fact, I have never really thought about what it looks like. It's fundamental group is the commutator subgroup of the fundamental group of C, which I believe is a free group of infinite rank. I don't think the field of meromorphic functions on this guy is what you want.
2) On the other hand, the Hilbert class group $G$ of $K$ can be viewed as the Picard group of $\mathbb{Z}_K$, which classifies line bundles on $\mathbb{Z}_K$. This generalizes nicely: the Picard group of $C$ is an exension of $\mathbb{Z}$ by a $g$-dimensional complex torus $J(C)$, which has exactly the same abelian fundamental group as $C$ does: indeed their first homology groups are canonically isomorphic. $J(C)$ is called the Jacobian of $C$.
3) It is known that every finite unramified abelian covering of $C$ arises by pulling back an isogeny from $J(C)$.
So there are reasonable claims for calling either $G \cong \mathbb{Z}^{2g}$ and $J(C)$ the Hilbert class group of $C$. These two groups are -- canonically, though I didn't explain why -- Pontrjagin dual to each other, whereas a finite abelian group is (non-canonically) self-Pontrjagin dual. [This suggests I may have done something slightly wrong above.]
As to what the Hilbert class field should be, the analogy doesn't seem so precise. Proceeding most literally you might take the direct limit of the function fields of all of the unramified abelian extensions of $C$, but that doesn't look like such a nice field.
Finally, let me note that things work out much more closely if you replace $\mathbb{C}$ with a finite field $\mathbb{F}_q$. Then the Hilbert class field of the function field of that curve is a finite abelian extension field whose Galois group is isomorphic to $J(C)(\mathbb{F}_q)$, the (finite!) group of $\mathbb{F}_q$-rational points on the Jacobian.
Here is a case where it is non-Abelian. I use $K$ of class number 3. If I use the Gross curve, it is Abelian. If I twist in $Q(\sqrt{-15})$, it is Abelian for every one I tried, maybe because it is one class per genus. My comments are not from an expert.
> K<s>:=QuadraticField(-23);
> jinv:=jInvariant((1+Sqrt(RealField(200)!-23))/2);
> jrel:=PowerRelation(jinv,3 : Al:="LLL");
> Kj<j>:=ext<K|jrel>;
> E:=EllipticCurve([-3*j/(j-1728),-2*j/(j-1728)]);
> HasComplexMultiplication(E);
true -23
> c4, c6 := Explode(cInvariants(E)); // random twist with this j
> f:=Polynomial([-c6/864,-c4/48,0,1]);
> poly:=DivisionPolynomial(E,3); // Linear x Linear x Quadratic
> R:=Roots(poly);
> Kj2:=ext<Kj|Polynomial([-Evaluate(f,R[1][1]),0,1])>;
> KK:=ext<Kj2|Polynomial([-Evaluate(f,R[2][1]),0,1])>;
> assert #DivisionPoints(ChangeRing(E,KK)!0,3) eq 3^2; // all E[3] here
> f:=Factorization(ChangeRing(DefiningPolynomial(AbsoluteField(KK)),K))[1][1];
> GaloisGroup(f); /* not immediate to compute */
Permutation group acting on a set of cardinality 12
Order = 48 = 2^4 * 3
> IsAbelian($1);
false
This group has $A_4$ and $Z_2^4$ as normal subgroups, but I don't know it's name if any.
PS. 5-torsion is too long to compute most often.
Best Answer
As BCnrd says, the theorem you want is geometric class field theory. One version says that the abelianization of the fundamental group of a curve over an algebraically closed field is the fundamental group of its Jacobian. One can use this to derive class field theory for curves over finite fields. Over the complex numbers, this is a topological statement, since the Jacobian of a Riemann surface can be constructed by topological methods, such as $J(C)=H_1(C;\mathbb R/\mathbb Z)$. Also, consider Weil's construction of the Jacobian by using Riemann-Roch to recognize a high symmetric power of a curve as a $\mathbb C\mathbb P^n$ bundle over the Jacobian. The projective space bundle is probably not a topological invariant, but the symmetric power is and it already has the right fundamental group. That has an extensive topological generalization, the Dold-Thom theorem, that the homology of a reasonable space is the homotopy groups of its infinite symmetric power.
The key unifying ingredient is, as BCnrd says, the Jacobian, even though it is missing from both of your statements.