When is this even possible (even for a dense graphs) $|E| = \Theta (|V|^2)$ Wikipedia says that "a dense graph is a graph in which the number of edges is close to the maximal number of edges."  and "The maximum number of edges for an undirected graph is $|V|(|V|-1)/2$".  Then why do even use $ |E| = \Theta (|V|^2)$?, understand that $\Theta$ is the correct(tightest) bound in asymptomatic notation.  It seems to me that $|E| = \Theta (|V|^2)$ can never happen, so why do we use it?
You're absolutely right that $\Theta$ is the tightest asymptotic bound. But it's still asymptotic, and that means that we don't care about constant factors or terms of lower degree: when $n$ (or $v$ or whatever) gets big enough, the smaller terms become negligible.

In this case, $\frac{v(v-1)}{2} = \frac{1}{2} v^2 + \frac{-1}{2} v$. So removing the constant factors and lower terms leaves us with $v^2$.