Quantum geometry is known to influence the effective mass of single particles in multiband systems through the geometric properties of Bloch states. Here, we show that this connection extends beyond the one-body level. Starting from a multiband Hubbard model with on-site attractive interaction, we derive an exact effective-mass theorem for two-body bound states in vacuum and establish the corresponding geometric framework for Cooper pairs near the superconducting critical temperature within Gaussian fluctuation theory. We demonstrate that pair effective masses contain not only ``conventional'' contributions arising from band dispersion and band geometry, but also distinct geometric terms governed by quantum metrics defined on the manifolds of paired states, provided pairing is non-uniform across sublattices. This reveals a hierarchy linking band geometry, pair geometry, and Cooper-pair geometry. In the many-body problem, analytic continuation naturally leads to a non-Hermitian fluctuation kernel and a biorthogonal quantum geometry, yielding a generally complex Cooper-pair effective mass when collective modes overlap with the fermionic continuum. Exact numerical calculations for the sawtooth, Su-Schrieffer-Heeger, Hofstadter, and fluorite-like lattice models show that pair geometry can provide a quantitatively significant contribution to the effective mass of bound pairs. Our results identify pair quantum geometry as an essential ingredient governing bound-pair dynamics and multiband superconductivity beyond band geometry.