We investigate the self-assembly of amphiphilic nanocubes into finite-sized aggregates in equilibrium and under shear, using molecular dynamics (MD) simulations and kinetic Monte Carlo (KMC) calculations. These patchy nanoparticles combine both interaction and shape anisotropy, making them valuable models for studying folded proteins and DNA-functionalized nanoparticles. The nanocubes can self-assemble into various finite-sized aggregates ranging from rods to self-avoiding random walks, depending on the number and placement of the hydrophobic faces. Our study focuses on suspensions containing multi- and one-patch cubes, with their ratio systematically varied. When the binding energy is comparable to the thermal energy, the aggregates consist of only few cubes that spontaneously associate/dissociate. However, highly stable aggregates emerge when the binding energy exceeds the thermal energy. Generally, the mean aggregation number of the self-assembled clusters increases with the number of hydrophobic faces and decreases with increasing fraction of one-patch cubes. In sheared suspensions, the more frequent collisions between nanocube clusters lead to faster aggregation dynamics but also to smaller terminal steady-state mean cluster sizes. The results from the MD and KMC simulations are in excellent agreement for all investigated two-patch cases, whereas the three-patch cubes form systematically smaller clusters in the MD simulations compared to the KMC calculations due to finite-size effects and slow aggregation kinetics. By analyzing the rate kernels, we are able to identify the primary mechanisms responsible for (shear-induced) cluster growth and breakup. This understanding allows us to tune nanoparticle and process parameters to achieve desired cluster sizes and shapes.