The current study delves into SH-wave dynamics within a composite structure, comprising a viscoelastic functionally graded layer resting on a magneto-electro-elastic graded substrate with loosely bonded interface. A generalized equation for SH-wave propagation is formulated and numerically analyzed, considering an interface allowing for limited slip with a linear shear-slip correlation. One of the principal findings of this study highlights the significant influence of loosely bonded interface on wave propagation velocity. The propagation characteristics within the FGMEE foundation are estimated utilizing the Modified Bessel Function approach. The dispersion and attenuation characteristics of SH-waves have been analyzed and depicted graphically under both electrically-magnetically short and open conditions. To further enrich the analysis, numerical computations were executed through MATHEMATICA software. Incorporating interfacial bonding effects and magneto-electro-elastic coupling effects in SH wave analysis would provide a more realistic representation of wave interface interactions in functionally graded viscoelastic and magneto-electro-elastic structures, thereby enhancing the novelty and practical applicability of the study. A comparative investigation has been conducted to examine the situations related to the functional gradation of both mediums. In addition, diverse conditions have been graphically represented that illustrate the impact of the dissipation factor, the interfacial bonding parameter, and the viscosity of the viscoelastic layer on the phase velocity and attenuation characteristics of SH-waves, emphasizing a key aspect of this research.