I do not know whether there is any advantage to considering this problem in the ring of Eisenstein integers $\mathbb{Z}[\omega],$ where $\omega = e ^{\frac{2 \pi i}{3}},$ which is a PID. Then we have to ask when we can have $(x - \omega y)(x +\omega y)(x - \omega^{2}y )(x+\omega^{2}y)$ dividing $(xy- \omega)(xy +\omega)(xy- \omega^{2})(xy +\omega^{2})$ in $\mathbb{Z}[\omega],$ where $x,y$ are rational integers. I have not been able to pursue this to provide further insight myself, but someone else might.
Later remark: It is easy to check that the power of $1-\omega$ dividing both expressions is the same: it is $0$ if $3$ divides $xy,$ and $2$ if $3$ does not divided $xy.$ Hence we can omit the prime $1-\omega$ from our considerations, and we only need to worry about primes in $\mathbb{Z}[\omega]$ such that $N(\pi)$ is a rational prime congruent to $1$ (mod $3$). If $\pi$ is such a prime dividing the leftmost product, we note that $\pi$ divides exactly one of the terms in the rightmost product (and, in fact, $\pi$ also divides exactly one term in the leftmost product). This leads (if the required divisibility holds ) relatively easily to the observation (already made by the OP) that the leftmost expression divides $y^{12}-1$ (and/or $x^{12}-1,$ there is symmetry in $x$ and $y$), but it is unclear to me at present whether this viewpoint provides any more useful information.