In SAO 2 did Asuna break up with Kirito? I noticed that when introducing Shino to Asuna and Lisbeth, Kirito mentioned them as his nakama. Why is that? Why is it that Kirito didn't introduce Asuna as his girlfriend? Is this what is normally done by a Japanese couple? Or did they broke up?
There is no reason to think so. When Kirito introduced Shino to Asuna and Liz at the Dicey Cafe, the three of them were preparing to bring Shino back to her hometown, where she encountered the gunman she shot and killed as a young child, so that she could accept the consequences of her actions and overcome her guilt of having killed a man. One typically doesn't revisit the scene of a childhood trauma with just anyone, so Kirito, whom Shino already knew and trusted through GGO, called Asuna and Liz his nakama to tell Shino that he trusted them -- therefore she should too. It was more important to convey that Asuna and Liz were trustworthy rather than the fact that Asuna was his girlfriend.