Here is another perspective on the matter.

Observe that $2 \mathbb{Z}$ is a _proper_ one-dimensional submodule of the one-dimensional module $\mathbb{Z}$.

But with vector spaces, you can never have a $k$-dimensional subspace that is properly contained in another $k$-dimensional subspace.

So with modules, the lattice of submodules can have some interesting and rich structures, which cannot be present in the lattice of subspaces of a vector space.

Indeed, the structure of the lattice of submodules of $\mathbb{Z}$ -- all of which are one-dimensional, and where containment indicates divisibility -- contains a great deal of information about the theory of numbers.