$\newcommand{\X}{\mathcal X}\newcommand{\Y}{\mathcal Y}\newcommand{\si}{\sigma}\newcommand{\B}{\mathscr B}$Since $\X$ and $\Y$ are Polish spaces, the corresponding Borel $\si$-algebras $\B(\X)$ and $\B(\Y)$ are countably generated, so that the theory of $\psi$-irreducibility expounded in the Meyn and Tweedie book is applicable.

Let us reproduce here relevant definitions and facts from that book:

P. 89: A chain $X$ on $\X$ is $\psi$-irreducible if it is $\phi$-irreducible for some non-trivial measure $\phi$ on $\B(\X)$ -- in the sense for each $x\in\X$ and each $A\in\B(\X)$ such that $\phi(A)>0$ there is some integer $n\ge1$ such that $P^n(x,A)>0$. (The condition that $\phi$ be nontrivial seems to be missing in the definition of the $\psi$-irreducibility in the book.)

By Proposition 4.2.2 (p. 90 in the book), if a chain $X$ on $\X$ is $\psi$-irreducible, then there is a probability measure (say $\psi_X$) which is in a certain sense maximal among all non-trivial measures $\phi$ on $\B(\X)$ such that the chain is $\phi$-irreducible. (This terminology in the book may be a bit confusing indeed.)

P. 91: $\B^+(\X):=\{A\in\B(\X)\colon\psi_X(A)>0\}$.

P. 109: A set $C\in\B(\X)$ is called a small set if for some integer $m\ge1$, some non-trivial measure $\nu_m$ on $\B(\X)$, all $x\in C$, and all $B\in\B(\X)$ we have $P^m(x,B)\ge\nu_m(B)$; then the set $C$ is called $\nu_m$-small.

Let now $X$ and $Y$ be two independent chains on $\X$ and $\Y$, which are $\psi_X$- and $\psi_\Y$-irreducible, with kernels $P$ and $Q$, respectively.

Fix any $(x,y)\in\X\times\Y$ and any $(A,B)\in\B^+(\X)\times\B^+(\Y)$.
By Theorem 5.2.2 (p. 112 in the book),
\begin{equation*}
\exists k\ge1\ \exists C\subseteq A\ \text{s.t. } C\in\B^+(\X),\ C\text{ is $\nu_k$-small},\ \nu_k(C)>0.
\end{equation*}

By Theorem 5.4.4 (p. 120 in the book), the period $d_X$ of the chain $X$ is the greatest common divisor (gcd) of the set
\begin{equation*}
E_C:=\{n\ge1\colon C\text{ is $(h_n\nu_k)$-small for some $h_n>0$}\}.
\end{equation*}
Also, it is easy to see and noted in the book that the set $E_C$ is closed w.r. to the addition. Since the chain $X$ is aperiodic, we have $d_X=1$, and hence eventually (that is, for all large enough $n$) we have $n\in E_C$, that is, eventually $C$ is $(h_n\nu_k)$-small, so that
\begin{equation*}
\forall u\in C\ P^n(u,A)\ge h_n\nu_k(A)\ge h_n\nu_k(C).
\end{equation*}
Also, the condition $C\in\B^+(\X)$ implies $a:=P^j(x,C)>0$ for some $j\ge1$. So, eventually
\begin{equation*}
P^n(x,A)\ge\int_C P^j(x,du)P^{n-j}(u,A)\ge\int_C P^j(x,du)h_n\nu_k(C)
=ah_n\nu_k(C)>0.
\end{equation*}
Similarly, eventually $Q^n(y,B)>0$. Thus, eventually
\begin{equation*}
(P\otimes Q)^n((x,y),A\times B)=P^n(x,A)Q^n(y,B)>0. \tag{1}\label{1}
\end{equation*}

Inequality \eqref{1} represents a kind of limited version of the $\psi$-irreducibility for the product chain. Also, again by Theorem 5.4.4 in the book, \eqref{1} implies that the product chain is aperiodic.

Here is an idea how to get the true $\psi$-irreducibility for the product chain. For the fixed $(x,y)\in\X\times\Y$ and any $F\in\B(\X\times\Y)=\B(\X)\otimes\B(\Y)$, let
\begin{equation*}
U(F):=U((x,y),F):=\sum_{n=1}^\infty (P\otimes Q)^n((x,y),F),
\end{equation*}
so that $U$ is a (possibly infinite) measure on $\B(\X\times\Y)$. Then \eqref{1} means that the measure $\psi_X\otimes\psi_Y$ is "absolutely continuous" w.r. to $U$ on the algebra generated by the product sets $A\times B\in\B(\X)\times\B(\Y)$.

So, it "only" remains to show that this (perhaps with some, hopefully mild, additional conditions) implies the true absolute continuity of $\psi_X\otimes\psi_Y$ w.r. to $U$ on the $\si$-algebra $\B(\X)\otimes\B(\Y)$, generated by the product sets $A\times B\in\B(\X)\times\B(\Y)$.