Distributed swarm formation recognition requires each robot to independently classify the global formation using only local peer-to-peer communication, without access to global position information or a centralized coordinator. We present a fully distributed pipeline in which each robot computes a local feature vector from its one-hop communication neighborhood, passes it through a shared-weight graph neural network, and accumulates a final embedding through fixed-round gossip averaging. We evaluate three feature vector representations across three communication radii. Results demonstrate that structural features inspired by graphlets outperform distance features at sparse connectivity, while distance features dominate at high connectivity. Their concatenated feature vector performs well at all stages, demonstrating their successes as complementary. To our knowledge this is the first fully distributed formation classification problem of such manner.