The holographic encoding interpretation from the Bekenstein-Hawking entropy is due to it scaling with the area $S_{BH} = A/4$ of the surface, rather than the volume $V$. While I am aware that most quantum systems have volumetric entanglement entropy, I cannot really find any 3 dimensional result for the von Neumann entropy, which one should compare with $S_{BH}$. It does seem like they scale with the "volume" though, for instance the one dimensional XXZ Heisenberg chain and Hubbard model scale with the length of the chain. There seems to be a post A measure of entanglement which distinguishes between classical and quantum correlations which claims that this is true, but gives no reference.
The reason for the perhaps provocative question in the title is that for classical systems, such as the ideal gas one finds an entropy $$S= N \left(\ln V +\frac{3}{2}\ln\left(\frac{m U}{3 \pi^2}\right)+5\frac{1-\ln N}{2}\right),$$ which I cannot see how one would interpret as a volume law. Here $\hbar = 1$. It is extensive in the particle number $N$ however. If we look at the Bekenstein-Hawking formula for a Schwarzschild black hole, then we find an entropy which goes like mass squared. I would call such an entropy super extensive, since doubling the number of particles which would form our black hole, more than doubles the entropy.
All in all, I would just like to hear thoughts on how the area law is supposed to be counter-intuitive or even to correspond to a reduction in complexity based on comparing and interpreting other entropies.
Note: I am not looking for an explanation based on how a quantum system can overshoot this bound, hence the need for quantum gravity, e.g., Lecture 6 Holographic Principle of Hong Liu. I am well aware of such arguments.
I am specifically looking for an understanding from a volume and area law perspective in the von Neumann entropy if one exists.