This dataset accompanies the manuscript by Nathan C. Huang, Rachael S. Skye, and Julia Dshemuchadse, "Advancing the Self-Assembly of Binary Colloidal Crystals of Tetrahedra and Octahedra with Confinement," Molecular Systems Design & Engineering, in publication (2025). In this project, we used Monte Carlo simulations to study the assembly of binary systems of polyhedral colloidal nanoparticles at surfaces with spherical and flat-wall geometries and to examine the influence of confinement on the process and products of crystallization compared to the bulk. We found that crystals formed in confinement exhibit higher degrees of crystallinity and lower quantities of secondary-phase defects: the formation of well-ordered layers and shells appears to be promoted by flat walls and spherical container interfaces. This dataset includes all simulation trajectories used for analysis (in .gsd file format) and a README to assist with parsing the data. We hope that this dataset will be useful for future work studying the assembly of colloidal crystals and polyhedral particles.