im sure egyptian mythology has everything to do with it
OP: does this question really need to be asked? cats are more graceful and elegant, they enjoy cleanliness and comfort. although there are obviously exceptions, dogs tend to prefer a rough, playful environment, oftentimes the outdoors. cats lounge about and clean themselves, dogs sniff, chase their tail and eat their own poop. the similarities are not difficult to recognize.
Sytpg's comment on Egyptian culture does raise a question. And that is that apparently this application of sex roles to the two different animals has been around a long time.
"No ray of sunshine is ever lost, but the green which it awakes into existence needs time to sprout, and it is not always granted to the sower to see the harvest. All work that is worth anything is done in faith." - Albert Schweitzer
because most (officially noticed) men are behaving in an "extroverted" way, that is to say, they subordinate to something objective, like an authority, a war, work, politics and dogs are known for acceptance of hierarchy. men are associated with agency, but its an ironic thing, as agency is the slave of relation. it needs the other, because without that it can not make sense of itself. and after all, the agent position secretly strives to die into communion.
and cats are concerned with their own safety and needs, like a queen, knowing of their individual value (since they can create life), they are creative (which is "introverted") they wont waste themselves to an objective, because without them there could be no objective. that is what cats do. women are associated with communion, wich may sound extroverted, but again is ironic: communion is creating a unique layer transcendence, which is selfless in a way, thus the individual who assumes identification with this perspective does not feel relative (agent/extroverted), its the mother of relations and demands to be approached through a relative style (a male style, and extroverted style, the style of someone who only lives to feed the cat and caress it).