Cat's and Dog's weren't engineered to be genetically compatible, the races of the Milky Way galaxy were.
I offered that as a joke. But we can play with this. When most of the races of the Milky Way were engineered to be genetically compatible by our Odo-looking progenitors, it stands to reason that Earth-bound life should be compatible with each other when advanced races who arise from completely different genetic backgrounds are. Humans were not specifically genetically engineered... all life on Earth was engineered to give rise to an advanced intelligent race, where primates won that battle. Would an early Earth hominid and a comparable ancient Klingon beast be able to procreate? Does this mean that alien species do not become similar until after they reach a certain complexity? What about races that look like dogs and cats? Is it somehow more reasonable to believe that Catians might be able to mate with those dog-faced aliens from season one of TNG when they are from completely different planets and backgrounds than it is for cats and dogs of Earth, who arose relatively recently from the same genetic ancestor? Would humans be able to mate with the Voth just because we are both advanced species? If so, would a chimpanzee be able to mate with a chicken when humans and Voth developed from the relatively same respective parent species?