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We establish that two-dimensional dipolar quantum gases admit a universal description, i.e., their thermodynamic properties are independent of details of the interaction at short distances. The only relevant parameters are the dipole length as well as the scattering length of the combined short-range plus dipolar interaction potential. We derive adiabatic relations that link the change in the thermodynamic potentials with respect to the scattering length and the dipole length to a generalized Tan contact parameter and a new dipolar contact, which involves an integral of a short-distance regularized pair distribution function. These two quantities determine the scale anomaly in the difference between pressure and energy density and also the internal energy in the presence of a harmonic confinement. For a weak transverse confinement, configurations with attractive interactions appear, which lead to a density-wave instability beyond a critical strength of the dipolar interaction. We show that this instability essentially coincides with the onset of a roton minimum in the excitation spectrum and may be understood in terms of a quantum analog of the Hansen-Verlet criterion for freezing of a classical fluid.