Sam Shepard's Buried Child won the Pulitzer Prize in 1979 and set the tone for future meditations on the decay of rural America. The play explores themes of identity in several ways, from double identity to mistaken identity to loss of identity.
Halie and Dodge begin the play exploring their double identities: who they are and who they used to be. It's clear that their best days are behind them, but that does not stop Halie from seeking comfort in her memories. Dodge is more circumspect, hiding his active drinking from his wife, while Halie is hiding an affair with the local minister, adding to the theme of dual identities.
Mistaken identity is introduced both through false/idealized memories, and through the introduction of Vince, Tilden's son, and his girlfriend, Shelly. No one recognizes Vince as a member of the family until he loses all remnants of being an outsider and comes back to the house belligerent, drunk, climbing through a window. When Tilden discloses that he had a child with his own mother, it is clear that the story has been dismissed as part of Tilden's intellectual disability. However, through the introduction of Shelley as an outside observer, the family mythology is viewed through new eyes and even raises the question as to whether Vince is also Tilden and Halie's son.
Loss of identity is represented in the barren land, which has not produced enough crops to live on for some time. With the loss of the crop is a loss of purpose, allowing for all of the dysfunction that has taken over the family. The final loss of identity is represented in death, with Dodge quietly passing away on the couch as his family spins around him.