When you say that people fought like Kilkenny cats, you mean they fought valiantly to the bitter end, even if they are both destroyed. For instance, ''The two political parties fought like Kilkenny cats over the matter''
Usually said of someone who puts themselves down (similar to false modesty) in the hope that others will contradict them, and in the process, compliment them.
Sam: I'm no good at drawing!Judy: Nonsense! You're an excellent artist!Bob: Aw, he was just fishing for compliments.
(UK) If someone is trying to convince people to do or feel something without any hope of succeeding, they're flogging a dead horse.
This is used when someone is trying to raise interest in an issue that no-one supports anymore; beating a dead horse will not make it do any more work.