The phrase isn't related to the well-known antipathy between dogs and cats, which is exemplified in the phrase 'fight like cat and dog'. Nor is the phrase in any sense literal, that is, it doesn't record an incident where cats and dogs fell from the sky. Small creatures, of the size of frogs or fish, do occasionally get carried skywards in freak weather. Impromptu involuntary flight must also happen to dogs or cats from time to time, but there's no record of groups of them being scooped up in that way and causing this phrase to be coined. Not that we need to study English meteorological records for that - it's plainly implausible.