Genome-wide analysis has observed an excess of coincident single nucleotide polymorphisms (coSNPs) at human-chimpanzee orthologous positions, and suggested that this is due to cryptic variation in the mutation rate. While this phenomenon primarily corresponds with non-coding coSNPs, the situation in coding sequences remains unclear. Here we calculate the observed-to-expected ratio of coSNPs (coSNPO/E) to estimate the prevalence of human-chimpanzee coSNPs, and show that the excess of coSNPs is also present in coding regions. Intriguingly, coSNPO/E is much higher at zero-fold than at nonzero-fold degenerate sites; such a difference is due to an elevation of coSNPO/E at zero-fold degenerate sites, rather than a reduction at nonzero-fold degenerate ones. These trends are independent of chimpanzee subpopulation, population size, or sequencing techniques; and hold in broad generality across primates. We find that this discrepancy cannot fully explained by sequence contexts, shared ancestral polymorphisms, SNP density, and recombination rate, and that coSNPO/E in coding sequences is significantly influenced by purifying selection. We also show that selection and mutation rate affect coSNPO/E independently, and coSNPs tend to be less damaging and more correlated with human diseases than non-coSNPs. These suggest that coSNPs may represent a "signature" during primate protein evolution.