There is no classification theorem for abelian groups in general, as i asked here: (Why) Is there no analogue to the classification of finitely generated abelian groups for abelian groups?
However, every abelian group embeds into a divisible abelian group, and those have a quite simple classification. Another 2 cases of situations like this are that every (smooth) manifold embeds into $\mathbb R^n$ and that every small abelian category embeds into a category of modules over a ring. I'm curious for further examples of "classification up to embeding" theorems.
