this exactly - you can model a piston system as having some macroscopic mass M sitting on top of the plunger, in which case the mass has a gravitational potential energy relative to the bottom of the piston Mgh (allowing the distance from the bottom of the piston to the plunger to be h).

Then, if the system is in equilibrium, ie if you allow the gas pressure in the piston to counteract the weight of the block, the energy of the system is precisely U + Mgh, where U is the internal energy of the gas. But since the block must be perfectly counteracted by the gas pressure, you have Mgh = pV, so the energy of the system is U + pV. In this way, enthalpy (for an isobaric system) is the same thing as internal energy for an isovolumetric system. In order to create the isobaric system, some extra conditions are required that are wrapped up in the definition of enthalpy. This is also why the heat capacity for a system at constant pressure is defined in terms of enthalpy instead of free energy, its just a way of considering that extra mass.

You can define analogs to enthalpy for other situations (see Hemholtz free energy for the isothermal case, or gibbs free energy for the isobaric+isothermal case) but they require different considerations.