One thing that jumps out to me about your theory is the compatibility of cats and dogs with humans versus, say, a fish with humans. I feel domestication plays a large role in this, as humans domesticated these species for one reason or another (i.e., practicality). This then turns into refinement of a given subspecies, as we also know that certain dog breeds have been bred to look or behave a certain way (such as the facial structure of pugs, or any shepherding breed and what purpose they serve). We can see this through an experiment being held that's actively attempting to breed a domesticated fox - I'll leave a link to the video here
So, long story short, humans have actively played a role in shaping cats and dogs into the purpose they serve presently. What caused humans to attempt to do this in the first place, as well as when this would've happened historically speaking, I don't personally know, but would be open to hearing more about if anyone happens to know. Point being, I feel that the domestication of these species plays a large role in your explanation when it comes to compatibility of souls or the innate connection.
These are my thoughts at least, but overall I enjoyed reading your explanation.