It is known in the cases of some types of elliptic curves, the rational points form a group whose endomorphism ring can be isomorphic to the order of an imaginary quadratic extension.
I wonder how "special" is this fact, or perhaps, if this is a defining property of elliptic curves. Is it possible to construct other curves with such endomorphism ring structure? Is it possible to build a curve whose endomorphism ring is any given ring? For example, I am particularly curious about cubic orders.
Probably you are thinking about the endomorphisms of the elliptic curve itself, the group of rational points form a finitely generated abelian group, whose endomorphism ring is not of particular interest for as far as I know.
What is special about elliptic curves among all non-singular projective algebraic curves, is that they have a group structure, without which the endomorphisms (which are understood to preserve the neutral element) wouldn't actually form a ring.
The correct generalization is not to curves of genus other than 1, but to abelian varieties, which are projective abelian group varieties, of higher dimensions. In fact, every curve of genus $g\ge1$ has a functiorial embedding into an abelian variety, its Jacobian. Elliptic curves are special in that they are their own Jacobian.
For general, higher dimensional abelian varieties, we have something similar. In characteristic 0, the possible endomorphism rings, or rather the endomorphism algebra's, which are obtained by tensoring with $\mathbb Q$, are classified by Albert. For example, see this blog entry. The situation is more complex, but the direct analog of elliptic curves whose endomorphism ring is an order in a imaginary quadratic number field, is that of simple abelian varieties of Albert type IV, or, if you want to look at curves, curves whose Jacobian is such an abelian variety.