We present a novel variational derivation of the Maxwell-GLM system, which augments the original vacuum Maxwell equations via a generalized Lagrangian multiplier (GLM) approach by adding two supplementary acoustic subsystems and which was originally introduced by Munz et al. for purely numerical purposes to treat the divergence constraints of the magnetic and the electric field in the vacuum Maxwell equations within general purpose and non-structure-preserving numerical schemes for hyperbolic partial differential equations (PDEs). The mathematical properties of the model are: (i) its symmetric hyperbolicity, (ii) the extra conservation law for the total energy density and, most importantly, (iii) the very peculiar combination of the basic differential operators, since both curl-curl and div-grad combinations are mixed within this kind of system. A similar mixture of Maxwell-type and acoustic-type subsystems has recently been also forwarded by Buchman et al. in the context of a reformulation of the Einstein field equations of general relativity in terms of tetrads. This motivates our interest in this class of PDE, since the system is by itself very interesting from a mathematical point of view and can therefore serve as useful prototype system for the development of new structure-preserving numerical methods. Up to now, to the best of our knowledge, there exists neither a rigorous variational derivation of this class of hyperbolic PDE systems, nor do exactly energy-conserving and asymptotic-preserving (AP) schemes exist for them. The objectives of this paper are to derive the Maxwell-GLM system from an underlying variational principle, show its consistency with Hamiltonian mechanics and special relativity, extend it to the general nonlinear case and to develop new exactly energy-conserving and AP finite volume schemes for its discretization.