A systems can be formal even if the rules are not explicit. For example, take sauna culture in the south of Germany. Upon entering a German spa, an outsider must look around to pick up the rules: where are people wearing clothing? Where are they naked? What do you do with your towel? When must you be silent? What is with the little hat and shoes? What is the ceremony with the essential oils? When do you applaud? When can you leave? And so on. Usually this is all unwritten, but it's highly formal.
A surprising amount of social life is like this. Conversations over dinner may appear free-form, but there are often formal rules, lurking under the surface: are we playing “Wait Your Turn, Then Say Something Relevant and Interesting”? Or has this become a game of “Indignant Pile-On” or “Clarifying That Point”?
There's a reason that formal social systems are so common—the clearer the rules, the more scalable a social system is: it's easier for people to join and to leave, to predict the impact of their actions, to accept newcomers, and to navigate by their own goals and values, once they've picked up the vocabulary.
A social network is more formal than a chatroom. A town meeting is more formal than a wild party. (The least formal thing would be, maybe, a group of friends that are tripping, where almost anything could be said or done at any time, or a game with arbitrary and changing rules like Calvinball, Nomic, or MUDs.)