One possible alternative name for monoids with zero would be $\mathbb{F}_{1}$-algebras: just like $R$-algebras are monoids in $(\mathsf{Mod}_{R},\otimes_{R},R)$, monoids with zero are monoids in $(\mathsf{Mod}_{\mathbb{F}_{1}},\otimes_{\mathbb{F}_{1}},\mathbb{F}_{1})\mathbin{``="}(\mathsf{Sets}_*,\wedge,S^0)$.

Now, every object in $(\mathsf{Sets}_*,\wedge,S^0)$ comes with a natural diagonal morphism $X\to X\wedge X$ given by the composition $X\to X\times X\twoheadrightarrow X\times X/X\vee X$ and also a projection map $X\twoheadrightarrow S^0$.

Because of these maps, every monoid with zero $M$ satisfying $M^\times=M\setminus\{0\}$ will be an example of a "Hopf $\mathbb{F}_{1}$-algebra", a Hopf algebra object in $(\mathsf{Sets}_*,\wedge,S^0)$.

Not every such Hopf $\mathbb{F}_{1}$-algebra arises as such an $M$, but this seems like a notion that is quite close to what you are looking for.

One possible alternative name for monoids with zero would be $\mathbb{F}_{1}$-algebras: just like $R$-algebras are monoids in $(\mathsf{Mod}_{R},\otimes_{R},R)$, monoids with zero are monoids in $(\mathsf{Mod}_{\mathbb{F}_{1}},\otimes_{\mathbb{F}_{1}},\mathbb{F}_{1})\mathbin{``="}(\mathsf{Sets}_*,\wedge,S^0)$.

Now, every object in $(\mathsf{Sets}_*,\wedge,S^0)$ comes with a natural diagonal morphism $X\to X\wedge X$ given by the composition $X\to X\times X\twoheadrightarrow X\times X/X\vee X$ and also a projection map $X\twoheadrightarrow S^0$.

Because of these maps, every monoid with zero $M$ satisfying $M^\times=M\setminus\{0\}$ will be an example of a "Hopf $\mathbb{F}_{1}$-algebra", a Hopf algebra object in $(\mathsf{Sets}_*,\wedge,S^0)$.

Not every such Hopf $\mathbb{F}_{1}$-algebra arises as such an $M$, but this seems like a notion that is quite close to what you are looking for.

Every Hopf $\mathbb{F}_{1}$-algebra arises in this way because the categories of comonoids in $(\mathsf{Sets},\times,*)$ and $(\mathsf{Sets}_*,\wedge,S^0)$ are equivalent, see Lemma 2.4 here. So your notion is equivalent to that of a Hopf $\mathbb{F}_{1}$-algebra, i.e. a Hopf algebra object in $(\mathsf{Sets}_*,\wedge,S^0)$.