Cell adhesion with extracellular matrix depends on the collective behaviors of a large number of receptor-ligand bonds at the compliant cell-matrix interface. While most biological tissues and structures, including cells and extracellular matrices, exhibit strongly anisotropic material properties, existing studies on molecular adhesion via receptor-ligand bonds have been largely limited to isotropic materials. Here the effects of transverse isotropy, a common form of material anisotropy in biological systems, in modulating the adhesion behavior of a cluster of receptor-ligand bonds are investigated. The results provide a theoretical basis to understand cell adhesion on anisotropic extracellular matrices and to explore the possibility of controlling cell adhesion via anisotropy design in material properties. The combined analysis and simulations show that the orientation of material anisotropy strongly affects the apparent softness felt by the adhesive bonds, thereby altering their ensemble lifetime by several orders of magnitude. An implication of this study is that distinct cellular behaviors can be achieved through remodeling of material anisotropy in either extracellular matrix or cytoskeleton. Comparison between different loading conditions, together with the effects of material anisotropy, yields a rich array of out-of-equilibrium behaviors in the molecular interaction between reactant-bearing soft surfaces, with important implications on the mechanosensitivity of cells.