Information geometric regularization (IGR) was recently introduced as the first inviscid regularization of the multidimensional compressible Euler equations, enabling large-scale simulations of realistic compressible flows. While IGR has demonstrated strong computational performance, the thermodynamic structure induced by this regularization has not yet been fully understood. In this work, we investigate the thermodynamic properties of IGR by decomposing its dynamics into a conservative Hamiltonian subsystem and a dissipative component. This decomposition motivates the introduction of two related models for comparison. The first is the Hamiltonian Regularized Euler (HRE) model, which constitutes the first multidimensional, non-dispersive Hamiltonian regularization of the compressible Euler equations with energy. The second is the Hamiltonian IGR (HIGR) model, which reformulates the dissipation in IGR within the metriplectic framework. Numerical experiments on colliding shock problems show that, despite their appealing structural properties, both HRE and HIGR exhibit significant deficiencies that limit their immediate use as computational models and point to the need for a deeper theory of dissipative weak solutions. Alongside these comparisons, we derive new analytical results for the IGR model itself, including exact conservation of acoustic waves, local energy transport laws, and explicit entropy production rates. By clearly separating conservative and dissipative effects, this framework provides a principled thermodynamic interpretation of IGR and establishes a foundation for future analytical developments and model extensions for inviscid regularizations of multidimensional compressible flows with thermodynamics.