Is there an intuitive reason that degeneracy pressure is intensive?

Semiclassically, quantum particles in $n$ dimensions take up a volume $h^n$ of phase space. Therefore, if you keep the position-space density constant, the volume of momentum space per particle stays the same. That means that the characteristic momenta of the particles is intrinsic. Since pressure goes with energy density, this implies the pressure is intrinsic as well.

First of all, I don't think this is true in general. Specifically, for systems with sufficiently long-range interactions, like $V\sim r^{-d}$ or longer range, the Fermi pressure will no longer be intensive. Indeed, the energy density will not be intensive for any particle statistics, and as a result the thermodynamic limit does not exist for such systems.

So a more general form of this question would be to ask: why does the thermodynamic limit exist for non-interacting particles? But I guess it is pretty much the definition of "non-interacting" that the particles don't change their energies when you bring several of them together!