I'll go ahead and turn my comment into an answer. It does form a monad, but (probably) not a very interesting one. Namely, first note that any pair of adjoint functors $L:\mathcal{C}\leftrightarrows \mathcal{D}:R$ is associated to a monad $RL$ on $\mathcal{C}$. Multiplication is given by the map $RLRL\overset{LR\to 1}{\to} RL$ and unit is $1\to RL$. Here the natural transformations $1\to RL$ and $LR\to 1$ (with $1$ denoting the identity functor) are the unit and counit of the adjunction, adjoint to the identity maps $L\to L, \text{ } R\to R$, respectively. Further, given an object $A$ of $\mathcal{D}$, we get an algebra $R(A)$ over the monad $RL$ with action morphism $RLR(A)\overset{LR\to 1}{\to} R(A)$ induced from the counit. This gives us a functor $\alpha:\mathcal{D}\to Alg(RL)$ to algebras over the monad $RL$. In "most real-life situations" where the functor $R$ is "forgetful" and $L$ is "free", the functor $\alpha$ to algebras is in fact an equivalence of categories. This can be checked formally, using something called the Bar-Beck monoidicity theorem (in a sense this is similar to the condition for an abelian category $\mathcal{A}$ to be the catgeory of modules over an algebra: in fact, modules over an algebra are a special case of algebras over a monad).

The monad you are considering is associated to the adjunction $A:Sp_*\leftrightarrows Ab:F$, where $F:Ab\to Sp_*$ is the forgetful functor from abelian varieties to pointed spaces and $A:Sp_*\to Ab$ is the Albanese functor. The monad is then $FA:Sp_*\to Sp_*$. The reason this monad is not very interesting is that the forgetful functor $F$ is fully faithful (this result in itself *is* very interesting, and encapsulates the rigidity of abelian varieties in algebraic geometry). This implies that the composition $FA:Sp_*\to Sp_*$ is an idemopotent functor. In particular, here we can apply the theory of idempotent monads, see e.g. https://ncatlab.org/nlab/show/idempotent+monad. Part 6 of definition 2.1 in the above notes then gives us for free that the adjunction $A:Sp_*\leftrightarrows Ab:F$ is monadic, i.e. the category $Ab$ of Abelian varieties is indeed equivalent to the category of algebras over the monad $FA$.