Although—as I have explained here—the phrase to rain cats and dogs is based on a cat-and-dog fight as a metaphor for a storm or hard rain, it is worth mentioning B. A. Phythian’s theory in A Concise Dictionary of Phrase and Fable (1993).

According to him, the phrase, or at least the metaphor, was in use for a long period before it was first recorded, and was originally referring to a disaster. He explains that a clue as to the origin of this phrase

is to be found in a quotation from Chaloner’s translation of Erasmus’ In Praise of Folly (1549):
         “Rather should we let all the world go to wreck both with dog and cat (as they say).”
This indicates that there existed a popular expression with dog and cat, that it was used of a disaster, and that it meant completely and utterly, down to the last dog and cat.

The praise of Folie (1549), the translation by the English statesman and poet Thomas Chaloner (1521-65) of Moriae encomium, written in Latin in 1509 by the Dutch humanist and scholar Desiderius Erasmus (circa 1469-1536), contains this passage:

Lesse sinne is it, (saie they) to slea a thousande men, than ones on a sondaie to clowte a poore mans shoe. Or, Rather shoulde we let all the worlde goe to wreke bothe with dogge and catte (as thei saie) than ones to make a lesyng, be the mattier never so lyght.

Bothe with dogge and catte is not a literal translation of the original Latin text, which is:

Levius esse crimen, homines mille iugulare, quam semel in die dominico calceum pauperi consuere ; et : potius esse committendum, ut universus orbis pereat una cum victu et vestitu, quod aiunt, suo, quam unicum quantumlibet leve mendaciolum dicere.

The literal translation of Erasmus’s Latin text is:

It is a lesser crime to kill a thousand men than to set a stitch on a poor man’s shoe on a Sabbath day; and a man should rather choose that the whole world should perish with all food and raiment,  as they say, than tell a lie, though never so inconsiderable.

So, the original text is una cum victu et vestitu, which translates literally as with all food and raiment.

This was presumably some kind of idiomatic phrase, and in search of an adequate translation, Chaloner chose bothe with dogge and catte.

And the most probable reason for this is that there really used to be a popular expression with dog and cat, and that it meant completely and utterly.

A heavy rain could have evoked the biblical story from the book of Genesis, in which God sent a great deluge from the sky to destroy the world. And the whole world would have gone to wreck both with dog and cat, had it not been for the righteous Noah and his Ark.

It is raining cats and dogs might thus have been motivated by the idea that it is raining as if it was the end of the world.