This article deals with the problem of output regulation in a "nonequilibrium" context for a special class of multivariable nonlinear systems stabilizable by high-gain feedback. A postprocessing internal model design suitable for the multivariable nature of the system, which might have more inputs than regulation errors, is proposed. Uncertainties in the system and exosystem are dealt with by assuming that the ideal steady-state input belongs to a certain "class of signals," by which an appropriate model set for the internal model can be derived. The adaptation mechanism for the internal model is then cast as an identification problem, and a least-squares solution is specifically developed. In line with recent developments in the field, the vision that emerges from this article is that approximate, possibly asymptotic, regulation is the appropriate way of approaching the problem in a multivariable and uncertain context. New insights about the use of identification tools in the design of adaptive internal models are also presented.