(as a sidenote on terminology, as Jonathan points out in the comments, "spectrum" is not really a good name for your objects precisely because you expect them not to be spectra)
If you look at theorem 8.8 in the GGN paper you cited, you'll see that categories with a nicely behaved tensor product yield examples by taking their groupoid core.
Here are some explicit classical examples:
For a commutative ring spectrum, $\mathrm{Proj}_R^\simeq$ is a commutative semiring space, whose ring completion is $K(R)$ when $R$ is connective. More generally, if $R$ is $E_k$, this semiring space "is" $E_{k-1}$.
There is an equivariant version of $\mathbb F$, namely $\mathrm{Fin}_G^\simeq$ for a (pro)finite group $G$; as well as an equivariant version of the previous example, where for a commutative ring $G$-spectrum $R$, you call "projective" any $R$-module which is a summand of a direct sum of $R\otimes G/H_+$'s (no shifts allowed)
Here's an example which doesn't directly fit into the context of [Thm 8.8, GGN] (although it can be made to, via e.g. the condensed/pyknotic approach):
$\mathrm{Vect}_\mathbb R^\simeq$, $\mathrm{Vect}_\mathbb C^\simeq$, the groupoids of finite dimensional vector spaces over $\mathbb{R,C}$ respectively, but where you take into account the topology on the hom-sets. Note that this is not literally $(\mathrm{Vect}_\mathbb K)^\simeq$ for some $\infty$-category of vector spaces and all linear maps obtained by applying the nerve to the topological category - indeed, if you do that then all mapping spaces are contractible and so you get trivial categories; so you have to restrict to isomorphisms before passing to $\infty$-categories. Their ring completions are $\mathrm{ko, ku}$ respectively.
Again, the above example has an equivariant version.
(the equivariant versions can be made into semiring $G$-spaces, but let me not get into that here)
There are probably tons of other examples.
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 $H$. 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 the homomorphism $U(B) \odot A \to A \otimes B$ whose component at an element $b \in B$ is $-\otimes b \colon A \to A \otimes B$.
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 $\mathcal{V} \to \operatorname{Fun}(\mathcal{C},\mathcal{C})$.
Best Answer
Of course the real question is whether abelian groups are really more fundamental objects than commutative monoids. In a sense, the answer is obviously no: the definition of commutative monoid is simpler and admits alternative descriptions such as the one I give here. The latter description can be adapted to other settings, such as to the 2-category of locally presentable categories, which shares many formal properties with the category of commutative monoids (such as being closed symmetric monoidal, having a zero object, having biproducts). As such I would claim that any locally presentable closed symmetric monoidal category is itself a categorified version of a semiring, not in the sense you describe, but in that it is an algebra object in a closed symmetric monoidal category, so we may talk of modules over it, etc.
However, it is undeniable that there is a large qualitative difference between the theories of abelian groups and commutative monoids. Observe that an abelian group is just a commutative monoid which is a module over $\mathbb{Z}$ (more precisely a commutative monoid has either a unique structure of $\mathbb{Z}$-module, if it has additive inverses, and no structure of $\mathbb{Z}$-module otherwise). The situation is analogous to the (smaller) difference between abelian groups and $\mathbb{Q}$-vector spaces. I do not know of a characterization of $\mathbb{Z}$ as a commutative monoid that can be transported to other settings. It seems that there is something deep about the fact that $\mathbb{Z}$-modules are so much nicer than commutative monoids, which often is taken for granted.