Is Jay Gatsby true to his identity

No.

Explanation:

His real name is James Gatz and he came from a no-name family in North Dakota. This is something he is quite ashamed of and so he creates a new identity for himself. Although he does gain wealth, he will never be like Tom and Daisy who were raised in that world. This is demonstrated through the extravagant car he drives and his attempt at imitating the lavish life he desires. It's absolute phony and characters see through this charade. He also tries so hard to relive the past that he abandons his current self. Gatsby truly lives in a fantasy world and thus, cannot be true to his identity.