Did Shakespeare really understand how universal his plays were in defining the many shades of the human experience?
That is a difficult question to answer definitively. Shakespeare was certainly a careful student of human nature, and his studies are reflected very eloquently in his explorations of human frailties in plays like Hamlet and Othello. In addition, he was a keen observer of the world around him, and he incorporates his own observations concerning politics, family life, and religion (among others) into his work. It is this ability Shakespeare had that has allowed for his work to become so universal.
Whether Shakespeare knew all of this - or even part of this - in his own time is a different matter. As Shakespeare certainly knew what he was doing in his portrayal of characters, I would argue that he aspired to a depiction of the "many shades of human experience." Shakespeare's aspiration, however, would only come to fruition in the years following his death.