Is this two input discrete system linear?
To be linear you must have a function $f$ that satisfy:

$ f(a\overline {x_1} + b \overline{x_2}) = af(\overline{x_1})+ b f(\overline{x_2}) $

I define $ \overline{x}=\left [\begin{matrix}\overline{x}(1) \\ \overline{x}(2) \end{matrix} \right] $ the vector of your two inputs at instant $n$

For each $n$ you have
$ y=f(\overline x)=\overline{x}(1) \cdot \overline{x}(2) $

Substituting to linearity definition:
$ f(a\overline {x_1} + b \overline{x_2}) = [a\overline{x_1}(1)+b\overline{x_2}(1)] \cdot [a\overline{x_1}(2)+b\overline{x_2}(2)] $

That is not equal to
$ af(\overline{x_1})+ b f(\overline{x_2}) =  a\overline{x_1}(1)\overline{x_1}(2)+ b\overline{x_2}(1)\overline{x_2}(2)$

It means that this system is not linear