No, it isn't the same thing. Depersonalisation can be brought on by stress and anxiety, but you need A LOT of it. Usually, it is associated to a traumatising experience. You can also have both an anxiety disorder and a depersonalisation disorder at the same time.

Signs of anxiety include (I'm basing myself on the DSM-IV) excessive worrying, muscular pain, tension, sleep problems, irritability, etc., whereas depersonalisation (again according to the DSM) is when you have the impression that you are a spectator to your own life and body.

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Hope that helps!