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.
I'm not sure about the cultural reasoning behind it, but I can at least say, from a story point of view, they have not broken up, since
Later in the story, Asuna and Kirito walk to Asuna's house so Kirito can properly greet Asuna's parents, which is not something you would do when you've broken up :P
That's about as non-spoilery I can get, but I hope it helps as an answer...
You guys are over thinking it. Remember when asuna was trapped in the game, how kirito went crazy trying to get her back. How in the all the show their friends call them lovebirds, cause of how they acted in public, or how spoiler if you haven't watch all of it. How when Yugi died and they were sitting on the grass kirito said to asuna "I don't know what's going to happen tomorrow, but what I do know is that want to be with you" might not be what he really said, but it's close, also shino called them lovebirds to.
You guys are overthinking. Don't expect an anime to reach its full romance right off the bat. Asuna loves Kirito, as she states that Kirito gave her a home to come to. And Kirito lives for Asuna, in the SAO arc remember how he poured his heart out to Asuna, saying that she is everything to him, that he lives for Asuna, and that they both will protect one another.
I really don't mind season 3 and 4,
people saying Asuna won't be main character
she is already a main character, but the LN tell stories from Kirito's perspective, and he seems to be a trouble-sponge. In GGO Kirito said that he can't afford to die, he now has a reason to live, Asuna. If they were to finalize Kirito's and Asuna's wedding, the anime wouldn't have much drama anymore. I feel like their weeding should be saved for the last, and final episodes.
And I don't mind the new volumes, I just hope they still bring up their relationship in at least flashbacks or them missing one another, as those 2 are meant to be together.
I heard something happens to Kirito after he comes back from UWO (season 3) and that Asuna goea to rescue her prince inside UWO (season 4).
In one episode, Shino was laying on Kirito's lap as he was staring at her butt. In this scene she asks if there's anyone out there that wouldn't want to see them like that. Kirito doesn't respond except with a nervous smile.
He also picked up Shino from her school as if he were her boyfriend. He didn't introduce Asuna as his girlfriend when Shino met her, he flirts with Shino in front of Asuna, and he treats Asuna as if she's not special, like he would a girlfriend. This makes it look like Kirito has moved on and Asuna is the one holding on to this relationship.