Lyric discussion by Appers66 

Cover art for As I Sat Sadly by Her Side lyrics by Nick Cave and the Bad Seeds

I think the thing about the kitten is that it goes to the person who has the next thing to say. So she has it in her lap as she says her part, and it moves to him for his verse, and then goes to her again for her last verse. It's almost an indicator, showing the different retorts and who controls the debate.