Let's work with a specific example.

What is the tensor product $Z_a \otimes Z_b$?

It can be shown that this product is also a cyclic group:

$Z_a \otimes Z_b = Z_c$, where $c = gcd(a,b)$.

For $x \in Z_a$ and $y \in Z_b$, the tensor product also gives us a specific tensor:

$x \otimes y \in Z_a \otimes Z_b = Z_{gcd(a,b)}$.