Here’s one sort of such evidence:
If different species of primates are to share common descent, their genetic differences should be consistent with the biochemical expectations of genetic mutations. That is, rather than being designed and hand-picked differences created for some functional purpose, the genetic differences should reflect a process of accumulating mutations. They do.
Then there’s the prediction of the chromosome 2 fusion. Since humans stand out as having 2 fewer chromosomes than our most similar primate cousins, if we truly are genetically and genealogically related through common descent, there must have occurred some sort of mutational change that made us end up with fewer chromosomes. Mutations that result in fewer chromosomes are known, one of which is a telomere-to-telomere fusion. Telomere-to-telomere chromosomal fusions produce internal telomeres (telomeres are normally found at the ends of the chromosomes), and they result in chromosomes with multiple centromeres.
Given the similar chromosomal banding patterns among humans, chimps, gorillas, and so on, the expectation was (if we were to share common ancestry) that human chromosome 2 suffered a telomere-to-telomere fusion. Sequencing human chromosome 2 confirmed this prediction of common descent, as it has both the expected internal telomeres and 2 centromeres.
And then there’s the all the other genetic similarities such as shared pseudogenes (vitellogenin, for example), ERVs, fossils, and the nested hierarchy. All evidence for common descent of all primates including humans.