Colloidal or nanoparticle mobility under confinement is of central importance for a wide range of physical and biological processes. Here, we introduce a minimal model of particles in a hydrodynamic continuum to examine how particle shape and concentration affect the transport of particles in spherical confinement. Specifically, an immersed boundary-general geometry Ewald-like approach is adopted to simulate the dynamics of spheres and cylinders under the influence of short- and long-range fluctuating hydrodynamic interactions with appropriate non-slip conditions at the confining walls. An efficient O(N) parallel finite element algorithm is used, thereby allowing simulations at high concentrations, while a Chebyshev polynomial approximation is implemented in order to satisfy the fluctuation-dissipation theorem. A concentration-dependent anomalous diffusion is observed for suspended particles. It is found that introducing cylinders in a background of spheres, i.e., particles with a simple degree of anisotropy, has a pronounced influence on the structure and dynamics of the particles. First, increasing the fraction of cylinders induces a particle segregation effect, where spheres are pushed toward the wall and cylinders remain near the center of the cavity. This segregation leads to a lower mobility for the spheres relative to that encountered in a system of pure spheres at the same volume fraction. Second, the diffusive-to-anomalous transition and the degree of anomaly quantified by the power law exponent in the mean square displacement vs time relation both increase as the fraction of cylinders becomes larger. These findings are of relevance for studies of diffusion in the cytoplasm, where proteins exhibit a distribution of size and shapes that could lead to some of the effects identified in the simulations reported here.