We perform computational studies of repulsive, frictionless disks to investigate the development of stress anisotropy in mechanically stable (MS) packings at jamming onset. We focus on two protocols for generating MS packings at jamming onset: (1) isotropic compression and (2) applied simple or pure shear strain gamma at fixed packing fraction phi. MS packings of frictionless disks occur as geometric families (i.e., quasiparabolic segments with positive curvature) in the phi-gamma plane. MS packings from protocol 1 populate parabolic segments with both signs of the slope, d phi/d gamma > 0 and d phi/d gamma < 0. In contrast, MS packings from protocol 2 populate segments with d phi/d gamma < 0 only. For both simple and pure shear, we derive a relationship between the stress anisotropy and local dilatancy d phi/d gamma obeyed by MS packings along geometrical families. We show that for MS packings prepared using isotropic compression, the stress anisotropy distribution is Gaussian centered at zero with a standard deviation that decreases with increasing system size. For shear jammed MS packings, the stress anisotropy distribution is a convolution of Weibull distributions that depend on strain, which has a nonzero average and standard deviation in the large-system limit. We also develop a framework to calculate the stress anisotropy distribution for packings generated via protocol 2 in terms of the stress anisotropy distribution for packings generated via protocol 1.