Is energy density and pressure fundamentally the same thing?

Yeah, where is the definition $ P=\frac{E}{V}$ from? It most definitely does not hold for all systems.

There are systems for which $ P=\frac{2E}{3V}$ (example: ideal classical gas) and systems for which $ P=\frac{E}{3V}$ (example: photon gas) and, generalizing these cases, systems for which $ P=\frac{sE}{3V}$ where the relation between energy and momentum is $E\propto p^{s}$ (independent of whether boson or fermions are in discussion).

So yes, they are closely related but they most definitely aren't one and the same thing.

From a statistical mechanics point of view, the energy density is really defined as:

$$ u=\frac{E}{v} $$

The pressure however is the conjugate variable of the volume, thus:

$$ P=\frac{\partial E}{\partial v} $$

The two are the same only when the energy is linear in the volume. This indeed may depend on the momentum as written above for some systems, But may also depend on other things. Think for example what happens when a piston is compressing gas.