Title: Kith and Kin
Summary: More than they are the Avengers, they are a family.
It is Natasha that is the first to come to the realization and acceptance that the Avengers are more than the Avengers. Tony is the one to suggest that they all live in the same building, yes, but it is Natasha that decides that the other are hers and starts acting accordingly. She quickly becomes the mother-sister of the team and like the spider she is named after, pulls those she cares about into the fold. It is not just her boys but also the ladies as well—everyone that she decides is part of her family becomes so. She is the most feared in their small family simply because she takes up that role because that means that the blackmail she has on them terrifies them all. The assassin isn't the strongest in the way the rest of the world judges them by, but the underestimation works for her so they are content with letting them judge her in that way.
Clint, of course, is soon to follow because Natasha and Clint are a pair. If Natasha pulls the others in, then Clint keeps them there. He makes sure to have an answer to everything and is willing to help whoever needs it. His ability to look at things from a distance makes him the only one that isn't biased whenever dealing with their numerous problems. With his patience and his easy going nature and his ability to take things in stride, he also ends up with the one to deal with the media and the press. Which also makes no sense because he's Clint, but somehow it works.
Thor is already starting to think of them as worthy comrades in battle. The push to think of the Avengers as kin is not a very big one and although he pauses a bit when he first thinks of them as siblings, it is a very easy change. He is the older one and is intelligent and easy to get along with. If Clint makes sure to have an answer, Thor makes sure to ask the right questions to let people make their own. Everyone else thinks of Thor as a child due to being an alien, well, then it is their fault for letting Thor steamroll right over them. The demigod is a prince and keen in politics.
Tony is the one to extend the invitation to live in the Tower, having already tailored it for each individual, but it takes a while for him to think of them as family. He doesn't realize he already does when he wakes up one day and realizes that he has gotten used to everyone quirks and spends his days working around them and fitting their equipment to them and that he has allowed everyone in. Tony isn't any one thing because that means trying to understand all of the facets and generalize and the world knows that it's impossible because he's Tony Stark. The others are perfectly all right in letting the world think that.
Steve is the one that first thinks that the Avengers can become a family given enough time. He had the Howling Commandos, after all. It's just that he doesn't want o find a new family. He's still mourning his first. It isn't until he realizes that he has taken the role that Bucky and the others had taken up for him—that he is healthy and fed and taken care of—often without realizing it that Steve decides that he has mourned enough and focuses everything he has on his new family. With Natasha, he takes over the role as family maker and often has conversations with her about such.
Bruce doesn't particularly think of them as family as much as close friends he would do anything for. Bruce doesn't get along with people. Even before the accident he has always been a bit mean and has trouble focusing on people rather than science. Yes, they are his best friends, and yes, he would go against the world for them, but it takes the combined effort of the others to make him realize that they don't care that he has green anger management issues. He is their family, no matter what.
And Hulk? Hulk is the overemotional little brother with a penchant for smashing things. The others would be lying if they didn't sometimes wish that they had an excuse to destroy things. Besides, Steve, who actually got to do things like that notices that Hulk does have Bruce's intelligence. It's just hidden by the extreme emotional issues. If the others notice that Hulk looks to them for guidance when he thinks that they aren't paying attention to him, they don't make any comments.
So yes, they are family. With numerous members of brothers and sisters and siblings and a whole host of aunts and uncles and other related words spread throughout the world. And they'd never want to be anything else.