This is hardly a complete answer.

As you say, all normal operators $a \in B(\mathcal H)$ satisfy this equality. Moreover, so does any element $c = a\oplus b \in B(\mathcal H\oplus \mathcal H)$ where $a$ is normal and $\|b\| \leq \|a\|$. This is because
$$ \|c^*c + cc^*\| = \max\{\|a^*a + aa^*\|, \|b^*b + bb^*\|\} = 2\|a\| = 2\|c\|. $$

On the other hand, this is certainly not a general property. For instance, if $a = \left[\begin{matrix} 0 & 1\\ 0&0\end{matrix}\right]$ then $$\|a^*a + a^*a\| = \left\|\left[\begin{matrix} 1 & 0\\ 0&1\end{matrix}\right]\right\| = 1 \quad \textrm{but} \quad 2\|a\|^2 = 2.$$

And on the third hand, $a = \left[\begin{matrix} 0 & 1 & 0 \\ 0& 0& 1\\ 0&0&0\end{matrix}\right]$ satisfies this equality. This makes me doubt that there will be any nice characterization.