You can understand this by looking at how they treat each other and what they mean to each other.
Think about Johnny, for example. His parents do not really like him. He gets no love or sense of belonging at home. Instead, he gets that stuff from the other members of the gang. The same goes for Dally, although he apparently doesn't get enough emotional support giving how he ends up. Speaking of Dally, look how far he goes to help Ponyboy and Johnny after Johnny kills the Soc -- it's like family.
So they are family because they care about each other and support each other the way a family is supposed to.
In the book "The Outsiders" the Greasers all know what it is like to be the outside group looking inward. They all have economic hardships, few own vehicles, and they have dysfunctional families. They also have a common enemy, the Socs. They are the kids who nobody seems to want or care about what happens to them. Pony Boy, Dally, and Soda Pop have each other and are a real family, but even they have lost their parents who died in a car accident. The boys provide each other with support, friendship, someone to get each other's back, a kind word as well as joking and laughter, and acceptance. In their homes many of them do not feel loved or have people who are concerned about them. However, among each other they are one and a part of them feels fulfilled where the gap of a family is lacking.
The greasers are like a family because they take care of each other like families take care of each other. One example is when Pony isn't feeling well and he confides in Two-Bit. Two bit continues to check his temperature and suggests he not fight in the rumble. Ponyboy convinces him that he will be fine and even says he'll take some aspirin as a prevention.