Examples of such sets $X$ can be generated knowing that $\|\lt u,v\gt\|_2 = \|u\|_2\|v\|_2\cos(u,v)$ and thus that if $\|v\|_2 = \frac{1}{\cos(u,v)}$, we have $\|\lt u,v\gt\|_2=\|u\|_2$.

if $u\in\mathbb{R}^1$ and $v=\left(\sqrt{s^2+t^2}\frac{1}{\sqrt{1+\frac{t}{s}}},\sqrt{s^2+t^2}\frac{\frac{t}{s}}{\sqrt{1+\frac{t}{s}}}\right)\in\mathbb{R}^2,s,t\in\mathbb{R}$, then $X:=\{v\}$ is a concrete example.

Examples of such sets $X$ can be generated knowing that $\|\lt u,v\gt\|_2 = \|u\|_2\|v\|_2\cos(u,v)$ and thus that if $\|v\|_2 = \frac{1}{\cos(u,v)}$, we have $\|\lt u,v\gt\|_2=\|u\|_2$.

if $u\in\mathbb{R}^1$ and $v=\left(\sqrt{s^2+t^2}\frac{1}{\sqrt{1+\frac{t}{s}}},\sqrt{s^2+t^2}\frac{\frac{t}{s}}{\sqrt{1+\frac{t}{s}}}\right)\in\mathbb{R}^2,s,t\in\mathbb{R}$, then $X:=\{v\}$ is a concrete example.

Edit:
for $u\in\mathbb{R}^1$ and $v\in\mathbb{R}^2$ we have
$\|v\|_2=\sqrt{x^2+y^2}$ and $cos(0,v)=\cos(\arctan(\frac{y}{x}))=\frac{abs(x)}{\sqrt{x^2+y^2}}$;
from $\|v\|_2=\frac{1}{\cos(0,v)}$ we get $\sqrt{x^2+y^2}=\frac{abs(x)}{\sqrt{x^2+y^2}}$ and finally, $y=\sqrt{abs(x)-x^2}$