The Zariski topology on an affine product is generally not the product topology

Its coordinate ring is $k[x]{\otimes }_{k}k[y]\cong k[x,y]$, so the diagonal is the algebraic set $V(x-y)$ and is Zariski closed.

Each factor has the cofinite Zariski topology. For a point $(a,b)$ off the diagonal, every basic product neighbourhood $U\times V$ of it has $U\cap V\mathrm{\ne }\mathrm{\varnothing }$, because both $U$ and $V$ are cofinite in the infinite field $k$. Choosing $c\in U\cap V$ gives $(c,c)\in (U\times V)\cap D$. Thus no product-topology neighbourhood of $(a,b)$ is contained in the complement of the diagonal.

Hence the diagonal is not product-topology closed although it is Zariski closed. The projections still have fibres obtained by quotienting $k[x,y]$ by the corresponding coordinate values, so the asserted fibre behaviour remains.