does the identity of molecules affect the internal energy, besides the temperature and volume?
By identity, I assume you mean for instance a carbon dioxide molecule versus a water molecule. If so, then yes, the stored energy depends quite strongly on identity. Much of chemistry is the prediction and understanding of the different molecular bond energies, because they determine if and how reactions take place.
If by identity you mean the same chemically, but different nuclear-wise, such as water versus heavy water, then the answer is still yes. There are different nuclear bonding states and thus different energies in the different elemental isotopes. Nuclear physics studies these different energies, and understanding them is crucial to design of nuclear weaponry and power plants.
If by identity you mean two water molecules with atoms of the same isotope, experiencing the same external environment, and having their electrons in the same configuration, then the answer becomes no. Beyond atomic type and arrangement, isotope configuration, and electron states, molecules do not have an identity. One water molecule is the same as the next.
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