Evidence of vortex rings abounds in inertial confinement fusion (ICF), supernovae, and, more generally, the Richtmyer-Meshkov instability (RMI). Besides classical piston-cylinder systems, vortex rings can emerge following the interaction of a shock with an interface separating two materials. Such rings have been shown to be a significant mechanism for the transport of circulation and energy in the RMI, with implications for turbulent transitions, and furthermore may be directly responsible for mixing that inhibits ignition in ICF and the ejection of heavy core elements in supernovae. Our objective is to examine the generation and scaling of vortex rings in compressible interfacial flows. We find that the circulation of such vortex rings ejected from shocked interfaces scales with the size of the perturbations from which they are generated up to a critical aspect ratio, analogous to the formation number of classical vortex rings, beyond which the properties of the rings saturate. We further investigate interactions between vortex rings and find that the structure of some supernova remnants can be explained by vortex core instabilities. Thus, an extensive body of classical literature and our extended theory can be used to inform the design of key engineering features in ICF targets and understand the specific mixing mechanisms responsible for the ejection of stellar core material in supernovae and the structure of their remnants.