The equation of state does not tell everything about a thermodynamic system. Moreover, the specific heat is not related to the value of the internal energy but to the variation of internal energy when the temperature changes.

A very simple example (even simpler than the case of interacting gases like Xenon and Helium) may help to understand the previous points.

Let's consider two equal volumes containing the same number of moles of two perfect gases at the same temperature. Gas A is made by monoatomic molecules, while gas B is made by di-atomic molecules. The equation of state is the same for both $PV=nRT$, therefore the pressure is the same. However specific heat and internal energy are not the same, being $U_A=\frac32 nRT$ and $U_B=\frac52 nRT$.

The reason for that has to do with the different roles of the energy of a single molecule in the internal energy and pressure. In this very simple case where all the energy is kinetic, the full energy (sum of the translational and rotational contributions) enters the thermodynamic internal energy. Only the translational degrees of freedom enter in the case of pressure, and this explains the equality of pressure.

In the case of interacting systems, the situation is even more complex but the main idea remains the same: the knowledge of the equation of state alone is not sufficient to reconstruct internal energy and specific heat. This is a basic fact of the description of any thermodynamic system.