I have no reason to support the idea that cats and dogs would take shelter in a thatched roof and, if the rain got hard enough, they could be washed out, leading to a lot of rain being referred to as "raining cats and dogs." But if you're thinking that this would require the cats and dogs to be ON TOP of the roof, you aren't picturing a thatched roof. A thatched roof is made of layers of rushes, reeds harvested from marshy land, laid over the rafters(?) in rows with the rushes tied in bundles. When the roof is completely covered in bundles of reeds, they cover it again with bundles of reeds. It isn't like a modern roof, a flat layer of thin shingles over a thin layer of felt; the thatching on a thatched roof would be a foot or more thick... and not as waterproof as you might think. But the thickness of the layers of thatch and the steep pitch of typical roofs allowed the water to run off the roof before it could rain into the house. If you picture such a roof in a rainstorm and, in your mind's eye, zoom in to a single drop of rain falling onto the roof, you will see it splash on a reed in the top layer of thatching. Some of it runs down that particular reed to the next one and on out of your sight. But part of that drop, and part of all the others striking that roof, ever, goes into the little valley where one reed meets its neighboring reed, and some of THAT water will leak through the very top layer of reeds, or thatching, to be intercepted by the next layer down and, mostly, diverted down toward the eaves and beyond the footprint of the house, as intended. But some of it will leak down through to the next layer, and so on. You can see why, if you're going to make a roof out of a bunch of skinny straws jammed up against each other and held in place by string, you would lay them on kind of thick.
Over time the roof would become quite the ecosystem, too, with lots of things living there. Basically every pest you would want to keep out of your house and lots of the the things that eat the pests and some of the things that eat the things that eat the pests would wind up living in, not on, the roof, making dens and pathways in the roof. This is another reason to make the roof extra thick, so it can keep keeping the rain out when half of it has been compromised by rodents building superhighways through it.
Of course, this roof is completely organic, so it's essentially compost. The rushes don't break down as quickly as lawn clippings by a long shot, but the process is the same over a 10- or 20-year span. So eventually you're sitting at your kitchen table, having a nice cuppa, and it's raining, of course, because it's always raining in Britain and Western Europe and that's where your thatched roof is so that's where your house is because it needs a roof, and it's about to need another one, because after 20 years the thatch really is compost, which is not a good construction material. In addition, today it is raining really hard so tons of water are falling on the roof every few minutes and all the beast and beasties that know enough to come in out of the rain, the insects and spiders and lizards and birds and mice and rats and yard cats and the dog you won't let in the house because he is so muddy, they all have retreated into the roof. Not on, in. When the roof fails in the downpour it goes from drips to streams to sagging to colllapsing in a hurry, and along with the water flooding down into your kitchen comes quite a lot of wildlife.
I suppose you could tell the neighbors that it was raining rats and lizards in your house, but that might reflect badly on you. Better, and just as truthful, to say it was raining cats and dogs.