In North America, you can find this story in the oral tradition of various indigenous peoples, but with a significant difference: the mix-up with the coats was intentional. Someone had a motive for setting fire to the house where the dogs were gathered.
When he asked, the dogs refused him.
You are unclean, they told Coyote,
you are not a dog.
So the dogs undressed for their sweatbath
and entered the sweat lodge without him.

"Why Dogs Smell Each Other's Butts," Lowell Jaeger
Yes, one of the native canids, Coyote, was there. But he was not invited inside to sweat with the dogs. He could have stolen their coats, then. But he didn't. He mixed up the pile of fur coats, muddied them so they were unrecognizable. And only then did he set fire to the roof, getting the dogs to race outside. Hearing Coyote's laughter, the dogs grabbed the first available coat they could find, thinking Coyote might have stolen some, and there might not be enough now to go around.
Years later, as the story goes,
with every dog zipped in someone else's fur,
dogs smell each other's butts,
looking for their own.

Both domesticated dogs, and the coyote (Canis latrans) have anal glands used to mark territory and identify individuals. The glands are located beneath the tails, so if you're a dog, and you need to check out another dog's scent, the anal/tail area is where you need to sniff.
Jaeger, Lowell. "Why Dogs Smell Each Other's Butts." Coyote's Journal. Berkeley: Wingbow Press, 1982.
MacCarter, Jane S. "The Coyote." Mountain Times. <> (25 February 2003)
"What are the Anal Glands?" 2001. <> (25 February 2003)
"Why Dogs Sniff Each Other's Butts." ShadowWoman's Forum. 7 July 1999. <> (25 February 2003)