Considering the predominant number of questions with either cats tag or dogs tag is it time for two separate sister sites - cats.stackexchange.com and dogs.stackexchange.com with this site only for non-cat and non-dog pets?
In addition to what James says, anything that would be on topic for CatsSE and DogsSE, would likely also be on topic here. So there is nothing gained by splitting up the sites that isn't already here. To draw a comparison, Stackoverflow gets a lot of Java questions, but there is no demand for a JavaSE, simply because all that would offer is already possible on SO.
Users that are only interested in cats/dogs/Java can already filter the homepage based on those tags. Users that are not interested in cats/dogs/Java can also filter the homepage to hide those questions. So this suggesting is trying to solve a problem that does not exist, while creating possible complications.
These complications could be for example that the other tags together don't generate enough traffic to keep the site above critical mass (I don't know if this is the case, but it could be true). If cats and dogs were to be split off, that might kill so much traffic to petsSE that the other tags "die." That would be a shame.
It is not going to happen.
New sites are proposed at https://area51.stackexchange.com/ go take a look at some of the recently closed sites. You will find close reasons like this
This subject is already well-covered by a live Stack Exchange site. We generally do not split off subjects simply to give them their own space.
The proposals don't last long, but there is a never ending series of spin off suggestions that are closed.