The example below shows that the bounds are tight.

Take the vectors $v=[1,2,\dots,(n-2),-\binom{n-1}2,0]$ and $w=[-1,-1,\dots,-1,(n-1)]$, then normalize them, with $\Vert v\Vert^2=\frac{3n-5}2\binom{n}3$ and $\Vert w\Vert^2=n(n-1)$, to get
$$f=\frac{v}{\Vert v\Vert}\qquad \text{and} \qquad
g=\frac{w}{\Vert w\Vert}.$$
Then, $\sum_{i,j}\max_{i,j}\{(f_i-f_j)^2,(g_i-g_j)^2\}$ equals
\begin{align} 2\sum_{1\leq i<j\leq n-2}\frac{(i-j)^2}{\Vert v\Vert^2}+
2\sum_{i=1}^{n-2}\frac{(i+\binom{n-1}2)^2}{\Vert v\Vert^2}+2\sum_{i=1}^{n-1}\frac{n^2}{\Vert w\Vert^2}=2(2n-1).
\end{align}

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