According to

Fajtlowicz, S.
Birkhoff's theorem in the category of non-indexed algebras.
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 1969 273-275.

a product of algebras and varieties was introduced by W. Narkiewicz. The terminology ``nonindexed product'' was used. It is stated in the above paper that the nonindexed product is the category-theoretic product in the
category of nonindexed algebras.

This is basically the thing that you are describing: the product of the clones of the varieties within the category of clones, or the product of their algebraic theories within the category of algebraic theories. Given varieties $\mathcal U$ and $\mathcal V$, the models of ${\mathcal U}\times {\mathcal V}$ are those algebras isomorphic to a set-theoretical product of some algebra $A\in \mathcal U$ with some algebra $B\in \mathcal V$, whose $n$-ary operations are pairs $(f,g)$ where $f$ is an $n$-ary operation of $A$ and $g$ is an $n$-ary operation of $B$. It is clear how these operations should act on $A\times B$, namely
$$
(f,g)((a_1,b_1),\ldots,(a_n,b_n)) = (f(a_1,\ldots,a_n),g(b_1,\ldots,b_n)).
$$

The conjecture that the product of the variety $\mathcal A$ of abelian groups with the variety $\mathcal M$ of monoids is the variety of monoids is not correct. There are two isotypes of monoids of size $2$, but there are $3$ isotypes of algebras in $\mathcal A\times \mathcal M$ that have size $2$.

Let me edit this to respond to comments: "isotype" = "isomorphism type". To expand on the preceding paragraph, let $A_2$ be the $2$-element group. Let $MA_2$ be the $2$-element monoid that is a group, and let $MS_2$ be the $2$-element monoid that is a semilattice. Let $*$ denote a $1$-element algebra of any type. Then the $3$ isotypes of $2$-element algebras in ${\mathcal A}\times {\mathcal M}$ are $A_2\times *$, $*\times MA_2$ and $*\times MS_2$. Observe that $A_2\times *$ and $*\times MA_2$ are not isomorphic, since there is a binary operation of ${\mathcal A}\times {\mathcal M}$ of the form $(f,g)(x,y)=(x+y,x)$. This agrees with the group operation on $A_2\times *$ but not on $*\times MA_2$.

Let me edit again to respond (slightly) to a question about identities satisfied by a product ${\mathcal U}\times {\mathcal V}$. Given varieties $\mathcal U$ and $\mathcal V$ and their identities, the identities of ${\mathcal U}\times {\mathcal V}$ have been worked out. I learned about a number of sources from Walter Taylor:

Taylor, Walter The fine spectrum of a variety. Algebra Universalis 5 (1975), no. 2, 263-303. [See Proposition 0.9]

McKenzie, Ralph
On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model.
J. Symbolic Logic 40 (1975), 186–196.

García, O. C.; Taylor, W. The lattice of interpretability types of varieties. Mem. Amer. Math. Soc. 50 (1984), no. 305, v+125 pp. [See Definition preceding Proposition 3.]

I won't list the details here, but will note that this issue (identities defining the product) has a complicated history. Taylor mentioned the names: Newman,
Foster, Pixley, Knoebel, Gratzer, Lakser, Plonka, Hu, Kelenson, Bernardi, Draskovicova, Chang, Jonsson, Tarski, Fajtlowicz, Lawvere, McKenzie, Garcia, Taylor.