The one line answer is that the category $\mathsf{Ab}$ of abelian groups is enriched over the **skew-monoidal category** $\mathsf{Gp}$ of groups, and that this "faux-tensor" defines a **skew-action** of the skew-monoidal category $\mathsf{Gp}$ on $\mathsf{Ab}$.

A *skew-monoidal structure* on a category $\mathcal{C}$ consists of a "tensor product" functor $\boxtimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, a "unit" object $I \in \mathcal{C}$, and "associativity and unit constraint" natural transformations $\alpha \colon (X \boxtimes Y) \boxtimes Z \to X \boxtimes (Y \boxtimes Z)$, $\lambda \colon I \boxtimes X \to X$, and $\rho \colon X \to X \boxtimes I$, satisfying the original five coherence axioms of Mac Lane. The important point is that these associativity and unit constraints are *not required to be invertible*. This notion was introduced by Szlachányi in his paper 

> Kornél Szlachányi. Skew-monoidal categories and bialgebroids. Adv. Math. 231 (2012), no. 3-4, 1694--1730. https://doi.org/10.1016/j.aim.2012.06.027

and has been much studied since, especially by the Australian school of category theory.

The "half-tensor products" of groups that you describe are part of a skew-monoidal structure on the category $\mathsf{Gp}$ of groups. This skew-monoidal structure is an instance of the family of examples described in Example 2.7 of my paper:

> Alexander Campbell. Skew-enriched categories. Applied Categorical Structures 26 (2018), no. 3, 597--615. https://doi.org/10.1007/s10485-017-9504-0

The tensor product $G \boxtimes H$ of two groups $G$ and $H$ is the group you denote by $G \triangleleft H$, i.e. the copower of $G$ by the underlying set of $H$, which is equally the direct product $G \times FU(H)$ of $G$ with the free group on the underlying set of $H$. Note that group homomorphisms $G \boxtimes H \to K$ correspond to functions $G \times H \to K$ that are group homomorphisms in the first variable. The unit object is the free group on one generator, i.e. $\mathbb{Z}$. The associativity and unit constraints are a little more complicated to describe, but suffice it to say that they are not invertible. 


This skew-monoidal structure on $\mathsf{Gp}$ is closed: the functor $- \boxtimes H$ has a right adjoint which sends a group $K$ to the group $[H,K]$ of all functions from $H$ to $K$ with the pointwise group structure; this group $[H,K]$ is the *internal hom* for this skew-monoidal structure on $\mathsf{Gp}$. Thus $\mathsf{Gp}$ is also a **skew-closed category** in the sense introduced by Ross Street in his paper:

> Ross Street. Skew-closed categories. J. Pure Appl. Algebra 217 (2013), no. 6, 973--988. https://doi.org/10.1016/j.jpaa.2012.09.020

Now, just as one can define categories enriched over monoidal categories, one can also define categories enriched over skew-monoidal categories. (In the terminology of my paper cited above, this is the same thing as a "left normal skew-enrichment" over the skew-monoidal category. Enrichment over skew-closed categories is defined in Street's paper cited above.)

We can define an enrichment of $\mathsf{Ab}$ over the above skew-monoidal structure on $\mathsf{Gp}$ as the change of base of the usual self-enrichment of $\mathsf{Ab}$ along the inclusion functor $\mathsf{Ab} \to \mathsf{Gp}$ equipped with the lax monoidal structure whose tensor constraint $A \boxtimes B \to A \otimes B$ is given by the composite $A \times FU(B) \to A \times B \to A \otimes B$ of $1_A \times \operatorname{ev}_B$ with the universal bilinear map $A \times B \to A \otimes B$. 

Unpacking this, we have that, for each pair of abelian groups $A$ and $B$, the hom-group $\underline{\operatorname{Hom}}(A,B)$ is the usual group of group homomorphisms from $A$ to $B$, with its pointwise group structure, but where we have forgotten that it's abelian. For each triple of abelian groups $A$, $B$, and $C$, the composition homomorphism $\underline{\operatorname{Hom}}(B,C) \boxtimes \underline{\operatorname{Hom}}(A,B) \to \underline{\operatorname{Hom}}(A,C)$ corresponds to the usual composition function $\operatorname{Hom}(B,C) \times \operatorname{Hom}(A,B) \to \operatorname{Hom}(A,C)$, but where we have forgetten that it's a group homomorphism in the second variable. Similarly, the unit homomorphisms $\mathbb{Z} \to \underline{\operatorname{Hom}}(A,A)$ simply pick out the identity homomomorphisms.

(Note that this enrichment of $\mathsf{Ab}$ over $\mathsf{Gp}$  can also be seen an instance of Example 2.7 of my paper cited above, since the category of abelian groups is equivalent to the category of group objects in $\mathsf{Gp}$.)

As you've spelled out in your question, the hom-functor $\underline{\operatorname{Hom}} \colon \mathsf{Ab}^\mathrm{op} \times \mathsf{Ab} \to \mathsf{Gp}$ is part of a two-variable adjunction, and so there are defined tensoring and cotensoring operations of an abelian group by a group. In particular, the tensoring operation defines a **skew-action** of the skew-monoidal category $\mathsf{Gp}$ on the category $\mathsf{Ab}$, in the sense of the paper:

> Stephen Lack and Ross Street. Skew-monoidal reflection and lifting theorems. Theory Appl. Categ. 30 (2015), Paper No. 28, 985--1000. http://tac.mta.ca/tac/volumes/30/28/30-28abs.html

Note that a skew-action of a skew-monoidal category $\mathcal{V}$ on a category $\mathcal{C}$ is simply an oplax monoidal functor $\mathcal{V} \to \operatorname{Fun}(\mathcal{C},\mathcal{C})$.