This question is inspired by $\mathbb{Z}_a\oplus\mathbb{Z}_b\cong \mathbb{Z}_c\oplus\mathbb{Z}_d$ question.
We change the additive structure to multiplicative:
Problem 1: If $a|b$, $c|d$ and $\mathbb{Z}^*_a \times \mathbb{Z}^*_b \cong \mathbb{Z}^*_c \times \mathbb{Z}^*_d$. Does this imply $(a,b)=(c,d)$?
The counter-example to Problem 1 is: $\mathbb{Z}^*_3 \times \mathbb{Z}^*_{12} \cong \mathbb{Z}^*_4 \times \mathbb{Z}^*_{12}$
And so I slightly modified it.
Problem 2: If $a|b$, $c|d$, $\mathbf{ab=cd}$ and $\mathbb{Z}^*_a \times \mathbb{Z}^*_b \cong \mathbb{Z}^*_c \times \mathbb{Z}^*_d$. Does this imply $(a,b)=(c,d)$?
Is there a counter-example now?