Up to isometry, how many product metrics are there?

Question

Given two metric spaces $(X,d_X)$ and $(Y,d_Y)$, you can form a product metric on the space $X \times Y$ by letting your metric be the $p$-product-metric for any $p \in [1,\infty)$ or the sup-metric for $p \to \infty$. But,

1. Do each of these metrics for $p \in [1,\infty]$ necessarily define isometric metric spaces? The blurb on Wikpedia only says this is true for Euclidian spaces^*.

2. What other metrics $d$ can we define on $X \times Y$ that are distinct up to isometry from these $p$-product metrics? Note that when defining a product metric $d$ on $X \times Y$, we should require that the restrictions of $d$ to $X$ and $Y$ make sense. Ie that $d|_{X\times\{y\}} = d_X$ for all $y \in Y$ and $d|_{\{x\}\times Y} = d_Y$ for all $x \in X$.