Does the absolute value of enthalpy have any physical significance?
The absolute value of enthalpy has direct relation to the mass of the system:

$$H = U + pV = mc^2 + pV$$

with the second term mostly negligible. If the system accepts thermal energy Q, its mass change is $\Delta m = \frac {Q}{c^2}$

For most of enthalpy applications, the absolute $H$ value is irrelevant.

The absolute value interpretation is the total energy stored in the system at the constant pressure, consisting of 


the equivalent energy to the mass of contained matter
work done by reaching the system volume against the external pressure.


This energy is formally extractable from the system by annihilation and shrinking to zero volume.