First, I seek a general characterization of 2-norm and 4-norm preserving matrices. Second, I seek to understand, using this characterization, why preserving the 2-norm can be described to have a far richer set of transformations than the 4-norm preserving ones.
I am trying to understand why quantum mechanics preserves only the 2-norm.
On
If I'm not mistaken, for real $a, b \ne 0$,
$$ |a + b|^p + |a-b|^p > 2 |a|^p + 2 |b|^p \ \text{if}\ p > 2$$ the inequality being reversed if $p < 2$.
As a consequence, if $e_j$ are the standard unit vectors and $T$ preserves the $p$-norm, the only way we can have $$\|T(e_i + e_j)\|^p_p + \|T(e_i - e_j)\|^p_p = \|e_i + e_j\|_p^p + \|e_i - e_j\|_p^p = 4 = 2 \|T(e_i)\|_p^p + 2 \|T(e_j)\|_p^p$$ is that there is no $k$ for which $(Te_i)_k$ and $(Te_j)_k$ are both nonzero. Thus the only real $n \times n$ matrices that preserve the $p$-norm are of the form $P D$ where $P$ is a permutation matrix and $D$ is diagonal with diagonal entries $\pm 1$.
As to why quantum mechanics uses the $L^2$ norm, that's a question of how the world happens to work, as far as we understand it...
Now mathematically, the $L^2$ norm has a huge advantage over any other $L^p$ norm: it is self-dual. In quantum mechanics, this manifests through the duality between bra and ket. To any state vector (bra) $|\phi\rangle$, you can aassociate a linear form (ket) $\langle \phi|$, and the nice property of the $L^2$ norm is that this preserves the norm. This is because the $L^2$ norm comes from a hermitian product, contrarily to the other $L^p$ norms.