**Theorem.** If $\mathcal V$ is a variety of finite signature and
$\mathcal V$ can be axiomatized by identities of the form $s\approx t$ where
$\textit{length}(s)=\textit{length}(t)$, then $\mathcal V$ is generated
by its finite members.

(For this theorem $\textit{length}(s)$ could be taken to be the number of symbols in $s$ written in, say, prefix notation. Or it could be the number of occurrences of variables and constants in $s$. Or it could be other things -- check the argument.)

_Proof._
Under the length hypothesis of the theorem, there is a congruence
$\theta_n$ on ${\mathbf F}_{\mathcal V}(m)$ which
relates two elements iff they are equal or else they can be
each represented by terms of length greater than $n$.
The signature is finite, so there are finitely many elements of
${\mathbf F}_{\mathcal V}(m)$ represented by terms of length at most $n$,
and therefore ${\mathbf F}_{\mathcal V}(m)/\theta_n$ is finite.
The intersection of the $\theta_n$'s is trivial, so this yields
a representation
$$
{\mathbf F}_{\mathcal V}(m)\hookrightarrow \prod_n 
{\mathbf F}_{\mathcal V}(m)/\theta_n
$$
of ${\mathbf F}_{\mathcal V}(m)$ as a subdirect product
of finite algebras. \\\

Your variety has finite signature, and the defining identity in prefix notation is $+*xy*yz \approx *+xy+yz$, with both sides of length $7$. Or, if we count length by the number of occurrences of variables and constants, then both sides have length $4$. In any case, the theorem shows that this variety
is generated by its finite members.