A reliable way to determine the effective properties of heterogeneous media plays an important role in engineering design. As an experimental test to obtain effective properties may be tedious and expensive to perform, and as computational approaches, when all heterogeneities of the media are considered during the simulations, might become impractical due to computational effort, homogenization procedures are a good alternative to estimate the effective properties of the media, mainly considering damage and manufacturing defects in the material. Among several homogenization methods, the Asymptotic Homogenization Method (AHM), a well-established mathematically based method, is considered for deriving the relations for the effective properties of the media, and the Finite Element Method (FEM) is used to solve the equilibrium relations. In this work, a fiber-reinforced composite is considered, with three different regions depicted: intact, damaged fibers, and regions with resin/void pockets. The proposed AHM-FEM approach is used to simulate a tensile test in all scenarios with standard and multiscale embedded Unit Cells used to obtain the effective elastic modulus on the fiber direction. Discussions regarding computational efficiency are carried out, as well as numerical comparisons among the types of Unit Cells used to model the media. In a second scenario, the complete fourth-order elasticity tensor of the media is investigated, considering the fibers as both isotropic and transversely isotropic. The results for the transversely isotropic fibers approach literature data, while when isotropic fibers are considered, poor results are achieved. Finally, the approach is extrapolated to obtain the mechanical properties of metamaterials, defined here as a composite reinforced by two different kinds of fibers.