Unconventional hydrocarbon reservoirs typically exhibit heterogeneity and anisotropy, resulting in complex elastic wave propagation characteristics. To improve understanding of elastic wave responses, this study investigates frequency-dependent anisotropy characteristics and develops an efficient time-splitting staggered-grid finite-difference algorithm for elastic wave propagation in transversely isotropic double-porosity media. Theoretical analysis indicates that global flow has a relatively weak effect on the dispersion and attenuation of fast waves (P1, S1, and S2), but it significantly affects two slow P waves (P2 and P3). P2 and P3 waves behave as diffusion modes at seismic frequencies but transition to propagation modes at ultrasonic frequencies or under low-viscosity conditions. Anisotropy has a strong effect on three P waves and two S waves. The proposed time-splitting staggered-grid finite-difference algorithm addresses the stiffness issue of poroelastic equations and promotes computational efficiency. Wavefield snapshots visually demonstrate the elliptical wavefronts of three P waves and the triplication phenomenon of S waves in anisotropic media. Fast-wave energy concentrates in the solid phase, while slow-wave energy is largely confined to the inclusion and background fluid phases, making it difficult to detect in reservoir exploration. The developed algorithm provides an effective numerical tool for simulating and analyzing wave propagation in transversely isotropic double-porosity reservoirs.