The sense of identity felt by Boy Willie and Berniece is focused on their family's accomplishments. This is expressed for Boy Willie in his pursuit of the land in Mississippi where his family was held as slaves. For Berniece, it's expressed in the piano itself.
Boy Willie connects to his past and his family's history by pursuing ownership of the land in Mississippi, where they were enslaved. However, he doesn't have enough money to purchase it. He insists on selling the piano that Berniece has had for years.
The piano was owned by the man who owned the Charles family. In fact, two members of the family were sold to pay for the piano. Papa Boy Willie—the husband of the woman sold—later carved the piano with scenes and portraits of the Charles family before they stole it from the slaveowners. Boy Charles died trying to get away from the slaveowner who was pursuing him for the theft.
To Boy Willie, the piano is just an object with sentimental value. To Berniece, it's something that her family created, fought, and died for. She sees it as disrespectful to their identity as the Charles family to get rid of it. When Boy Willie decides to sell it anyway, ghosts of the slaveowners and ghosts of the Charles family interfere, and he eventually comes around to see the piano as Berniece does—as a symbol of the family's history and identity.