Jobb, I believe a badger's head has a white spot on it which can make it look like the animal is bald. That may be the origin of the expression. As you know from the article, this headline is just saying that if you pet a cat affected by this disease (or a dog or other pet), you can catch the disease, and it could make you bald.
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The original term was “as bald as a badger’s backside”. In times gone by, barbers used shaving brushes, and brushes to dust cut hair off the customer, that were made from badger hair. The brush makers used to trap badgers and remove the hair from their backside then set them free to grow more for next time.