Now, can we picture these tensors _themselves_ as bilinear machines from $Z_a \times Z_b$ into the ground ring $Z$?

I don't think so.

We already saw that the only linear function from $Z_n$ into $Z$ was the zero function. Hence the only bilinear function from $Z_a \times Z_b$ into $Z$ is the zero function.

But there are $gcd(a,b)$ tensors in $Z_a \otimes Z_b$. So the tensors cannot be identified with the bilinear functions.

This is a basic picture of the tensor, as an object, which falls apart when we generalize from vector spaces to modules.

Or, have I made a mistake, and the picture actually be retained?