Fight like cats and dogs.

In the cartoon the husband and wife are fighting, but with words rather than physically. The dog has accused the cat of having her claws out, a different idiom meaning to fight like a cat. The cat accuses the dog of constantly yapping, again a different idiom meaning to sound like a dog barking all the time. 

To 'fight like cats and dogs' means to argue or fight intensely and constantly, verbally, physically or both. 

Let's see this idiom used in a conversation: 

Do you get along well with your brother?
Oh, yes. We're best friends.
That's great!
It hasn't always been like that though.
Really?
Sure. When we were kids, we fought like cats and dogs. 
Typical brothers!
I suppose so.