Let $G$ be a graph of order $n$ such that for each vertex $v$ there are two associated vectors, $f_v, g_v\in R^n$, where $uv\in E(G)$ if and only if $\|f_u - f_v\|^2 \ge \|g_u-g_v\|^2$.

ISGCI didn't discuss such a class.

Has this class of graphs been studied? What are their properties? Are they perhaps equivalent to some well-known graph class?

Note that, if the scalers is associated to vertices, from this post I know the answer is the class of permutation graphs. Now, considering vectors (in $R^\ell$ for some $\ell \ge 2$) is a natural genralization.