The French expression 'comme chien et chat' might date back to the 16th or 17th century. It means what you think it means: to fight like cat and dog. The strict translation is 'like dog and cat'. The English idiom is to fight like cat and dog whereas the French version is to be (in French 'comme' or like) a cat and dog.
The saying is based on the rather old-fashioned belief that cats and dogs always fight; not true in millions of households because the dog is socialised to cats and vice versa.
However, the dog chases the cat because dogs are predators. But the cat might chase a tiny dog as cats are predators too.
The English version might be used like this: "They've been married for 40 years but for the past ten, they fought like cat and dog".
Or "They love each other but fight like cat and dog."
It is often used in the context of a love-hate relationship.