Intracellular transport is supported by enzymes called motor proteins that are often coupled to the same cargo and function collectively. Recent experiments and theoretical advances have been able to explain certain behaviors of multiple motor systems by elucidating how unequal load sharing between coupled motors changes how they bind, step, and detach. However, non-mechanical interactions are typically overlooked despite several studies suggesting that microtubule-bound kinesins interact locally via short-range non-mechanical potentials. This work develops a new stochastic model to explore how these types of interactions influence multiple kinesin functions in addition to mechanical coupling. Non-mechanical interactions are assumed to affect kinesin mechanochemistry only when the motors are separated by less than three microtubule lattice sites, and it is shown that relatively weak interaction energies (~2 kBT) can have an appreciable influence over collective motor velocities and detachment rates. In agreement with optical trapping experiments on structurally-defined kinesin complexes, the model predicts that these effects primarily occur when cargos are transported against loads exceeding single-kinesin stalling forces. Overall, these results highlight the inter-dependent nature of factors influencing collective motor functions, namely, that the way the bound configuration of a multiple motor system evolves under load determines how local non-mechanical interactions influence motor cooperation.