Genetically speaking, are dogs exactly similar to humans and chimps both?
Dawkins is overgeneralizing.  Genetic distance is a fairly new field, and a heated topic, since your answer depends upon which genes you are comparing.  If you picked ONLY strongly-conserved genes for your genetic distance comparison, then you'd find that humans, bonobos, dogs, mice, mosquitos, shrimp, and bakers' yeast are all equally distant (and that the distance is small).  But if you pick ONLY very weakly-conserved genes for your genetic distance comparison, you can get crazy results -- humans are more closely related to yeast than to horses, for example.  (Creationists have had a heyday making fun of some early genetic distance studies that, in retrospect, had poorly-chosen genes.)  So biologists doing genetic distance comparisons try to pick genes that are common to many species, and that have some drift to them, but not so much that the answers are obviously goofy.  Another problem that has arisen is that sometimes an entire gene in several species appears to have been obtained from a very dissimilar species, in a different phylum or kingdom even.  (Current theory is that this is the work of viruses.)  The "tree of life" paradigm might need to be changed to the "web of life."  Needless to say, there's still a lot of argument about which (and how many) genes to pick for genetic distance studies.