We derive the conditions under which the fluid models obtained from the first two moments of Hamiltonian drift-kinetic systems of interest to plasma physics, preserve a Hamiltonian structure. The adopted procedure consists of determining closure relations that allow to truncate the Poisson bracket of the drift-kinetic system, expressed in terms of the moments, in such a way that the resulting operation is a Poisson bracket for functionals of the first two fluid moments. The analysis is carried out for a class of full drift-kinetic equations and also for drift-kinetic systems in which a splitting between an equilibrium distribution function and a perturbation is performed. In the former case we obtain that the only closure that leads to a Poisson bracket, without involving operators or an explicit dependence on the spatial coordinates, corresponds to that of an ideal adiabatic gas made of molecules possessing one degree of freedom. In the latter case, Hamiltonian closures turn out to be those in which the second moment is a linear combination of the first two moments, which can be seen as a linearization of the Hamiltonian closure of the full drift-kinetic case. A number of weakly-3D Hamiltonian-reduced fluid models of interest, for instance for tokamak plasmas, can be derived in this way and, viceversa given a fluid model with a Hamiltonian structure of a certain type, a parent Hamiltonian drift-kinetic model can then be identified. We make use of this correspondence to identify the drift-kinetic models from which Hamiltonian fluid models for magnetic reconnection and compressible plasma dynamics in the presence of a static but inhomogeneous magnetic field can be derived. The Casimir invariants of the Poisson brackets of the derived fluid models are also discussed. It is also shown that the Poisson structure for the fluid model derived from the full drift-kinetic system coincides with that of a reduced fluid model, when using the fluid velocity instead of the momentum as a dynamical variable.