This is not a complete answer, just a translation of Haskell to math so that the resident category theorists can tell what the question is.
As is usual in such cases, the terminology and the concepts in Haskell are slightly modified concepts from mathematics. It is best in the first iteration to look just at the broad similarities and not worry so much about the details.
Recall from Section 7 of the paper by Connor McBride and Ross Paterson that Haskell's Applicative is a lax monoidal functor (kind of as the Haskell definition uses internal Hom-sets where one would expect the external ones).
The definition of Alternative translates into: a functor $F : \mathcal{C} \to \mathcal{C}$, where $C$ is at least cartesian, together with natural transformations $e : 1 \to F$ and $m : F({-}) \times F({-}) \to F$, or concretely, for every object $A \in \mathcal{C}$ we have arrows
$$e_A : 1 \to F(A)$$
and
$$m_A : F(A) \times F(A) \to F(A)$$
such that $(F(A), e_A, m_A)$ is a monoid for all $A$, naturally in $A$. What would you call such a thing?
The next question to ask is whether there is anything extra about having a lax monoidal functor which also has the above structure of monoids? I don't know off the top of my head. If there is another view of the same situation, it will surely be useful for various Haskell hacking tricks. Haskell people are very good at using category-theoretic algebra for all sorts of cool purposes.
Let me also explain on the difference between parametricity and naturality. When we define a "functor" $F$ in Haskell, that is not really a functor. It is acertain mapping from types to types which acts on internal hom-sets as a functor would. In addition, this mapping has a strong property known as parametricity which says that $F$ behaves "in the same way" on all types. Parametricity is quite similar to naturality, and there is a way to view it semantically as such, but it is not precisely naturality. This is not a detail into which we should worry about here, though.