i believe you can never turn into another species, but the idea of converging species is worth thinking about im sure. As you may know evolution is a change of a species through each instance of of children. So if the situation encourages a certain change, subsequential generations will aproach that change. So if dog like traits are more practical, cats with dog like traits will tend to survive and do the sex to make babies, those babies carrying those traits most likely. So over time cats would begin to look like dogs, to the point the effectively act like one. sans the procreation.
what i think your asking is why dont dogs turn into cats or vise versa, yes? An understandable question, but one that entails you dont understand how evolution works, that s fine you are learning. a good question to ask. An organism will not change drastically to turn into another species. you are human and you will die human. you can adapt however and change what you can. But that change is artificial. that would let you turn a cat into a dog though, so its cool.
take this with a grain of salt. i study physics, not biology.