Man, I've been reading these for years, and honestly, the most common thing I see is the 'Aincrad Survivors' bond trope. It makes total sense—they both went through that nightmare, so writers latch onto that shared trauma as a foundation. A lot of fics explore them meeting up post-'SAO' and dealing with PTSD together, which can be really heavy but also feels earned. They'll have these quiet scenes where they don't even need to talk about the game, just sitting somewhere peaceful, and it says everything. Sometimes it gets a bit repetitive though, leaning too hard on the angst without letting their personalities actually bounce off each other.
Another pattern is the 'Mutual Understanding Beyond Words' thing. Because Kirito isn't exactly a chatterbox and Azuna has her own quiet strength, fics love to paint them as two people who just get each other instinctively. It's less about dramatic confessions and more about subtle gestures—sharing a meal, a glance across a room in 'ALO', that sort of quiet comfort. I think it works best when it's balanced with moments of normalcy, like them trying to figure out how to date in the real world after knowing each other in a life-or-death situation. The fics that nail that awkward, real transition are my favorites.