In almost all of Shakespeare's works, there are identity issues, with many of them related to other things read in Great Books of Literature.
King Lear
There are many characters with identity issues within King Lear, but the most obvious and important is the one of the King himself. He was just leading England and had ultimate power in the land and believed that nothing could possible go wrong, until his game. He ends up dividing his kingdom believing that his daughters, Goneril and Regan, would take care of him. Well, that is not what happens and here is where his crisis begins. Lear does not know how to handle not having any power anymore and essentially being a bum or hobo. He is no longer a King, but he keeps trying to act like he is one to no avail and this identity crisis eventually causes him to go insane.
Julius Caesar
The identity crisis in this amazing work of Shakespeare is not held by Caesar, but by his betrayer Brutus. Brutus has always been faithful to Julius Caesar and never wanted to do wrong to Caesar, but all that changes when the senators try to get him to go against Caesar. Brutus does not know what to do because he is loves Caesar, but he is told Caesar wants to turn Rome into a monarchy. His crisis is that he must chose between being a Roman or being a good friend to Caesar. He eventually goes against Caesar and faces another identity problem when he realizes that the senators lied to him. Eventually, Brutus commits suicide from the problems dealing with his identity.
A Midsummer Night's Dream
The identity crises in this play are probably the most evident than in any other work of Shakespeare. The reason being that Puck the fairy messes with everyone's love life and makes them fall in love with all the wrong people. The problems that arise from this are the fighting that occurs for the people they love and the confusion of what is going on. For example, Hermia loves Lysander, but Lysander gets the drop on his eyes and falls for Helena. Hermia then thinks that either she or Lysander has gone crazy and it just gets worse from there, until Puck fixes it all. The funniest problem would have to be Titania and Bottom, as the queen of the fairies falls for essentially a donkey.
Romeo and Juliet
The amazing tale of star crossed lovers deals with one of the most important identities that a person can have and that is their family. As most people know, the tale involves the Montagues and the Capulets and their ongoing family feud, but then Romeo and Juliet fall in love. Juliet says "What's in a name? That which we call a rose by any other name would smell as sweet." What she means is that a name essentially is useless and meaningless, especially when you are in love. The crisis is that they belong to certain families and do not if they should leave this identity behind and the feud that they are in. Eventually this identity problem causes the fateful ending where the lovers both kill themselves.
Twelfth Night
One of the confusing, in a way, identity crises in all of Shakespeare occurs in this play, between Viola and her twin brother Sebastian. Believing that her brother is dead, Viola decides to dress like a man and act like she is a eunuch. Her identity issue is dealing with all the love drama that is going on, including women falling for her, while not blowing her cover. The end is comical at Shakespeare's best when Sebastian comes back to see his sister and they all believe he is the eunuch that Viola was acting like she was.