Canonical metric on product of two complete metric spaces
You can define, for instance,$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\max\bigl\{d_X(x_1,x_2),d_Y(y_1,y_2)\bigr\}$$or$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=d_X(x_1,x_2)+d_Y(y_1,y_2),$$or even$$d\bigl((x_1,y_1),(x_2,y_2)\bigr)=\sqrt{d_X(x_1,x_2)^2+d_Y(y_1,y_2)^2}.$$Any of them will do. And they all induce the product topology.
But it is not true that $X\times Y$ is complete with respect to any distance that you define on it, even if it induces the product topology.