It's the tensor unit: for any module $V$ we have $V \otimes k \cong k \otimes V \cong V$ as $\mathfrak{g}$-modules.

There are cases in which you must take the tensor structure into account to distinguish the trivial module from the other simples. One example is a semisimple complex Lie algebra, whose module category is semisimple, so equivalent (as an abelian category, not as a tensor category) to the category of representations of a certain quiver with no arrows. Here we clearly can't tell the difference between simples without some extra structure.

As another example (not quite the same thing you are talking about, but related) consider representations of the *restricted* Lie algebra $\mathfrak{sl}_2$ in characteristic $p>2$. The module category of any non-semisimple block is equivalent to the category of representations of the algebra $kC_2 \rtimes k[x,y]/(x^2, y^2)$ where $C_2=\langle g \rangle$ is cyclic of order two, and $g$ conjugates each of $x$ and $y$ to its additive inverse. This algebra has two simple modules, each one-dimensional with $g$ acting as $\pm 1$. The algebra has an automorphism sending $g$ to $-g$ and fixing $x,y$, which lifts to an automorphism of the module category exchanging the two simple modules. Thus you can't tell the difference between them if you only know the structure of the category.

As usual there is a lot of insight to be had from the case of $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$. As an abelian category, $\mathfrak{g}$-$\operatorname{mod}$ is rather dull. As a tensor category it has a huge amount of interesting structure, involving Catalan numbers, the Temperley-Lieb algebras of statistical physics, "spiders"... -- see section 2 of Kuperberg's http://arxiv.org/pdf/q-alg/9712003v1.pdf for example.