Probably the most famous story from the Zhuangzi is the one in which he dreams so vividly that he is a butterfly that upon awakening he is unsure whether he is a butterfly now dreaming he is a man or was a man dreaming he was a butterfly. On the surface, this would seem to be intended to demonstrate that life is a dream. But though the fluidity of our perceived reality is here suggested, this simply makes possible something more central to Zhuangzi's philosophy, the fluidity of identity.
The story closes: "Surely, Zhou and a butterfly count as two distinct identities! Such is what we call the transformation of one thing into another." (Zhuangzi, 2:49; Ziporyn) The real question isn't whether Zhuangzi is a man or a butterfly, but how he can be both. But if he can be both, then might it not be better to say he is neither? And if "neither-of-the-two" (the literal meaning of "penumbra" in the preceding story), then he would seem to have no true identity at all. Identity, it seems, is not the fixed and sure thing we typically think it to be. Today he can be a butterfly, tomorrow a man, and the day after that...perhaps simply dispersed matter. If there is identity here, it is "peculiarly unfixed"; it is not identity as we know it, nor are we able to do so.
This is not Zhuangzi experiencing himself, his identity, in various forms; "these are two distinct identities". There is a discreet Zhuangzi identity, and a discreet butterfly identity. But identity, in this case, cannot be the fixed thing we generally take it to be. Every "fixed-identity" is lost in the transforming flow.
For this reason, we can say that we do not truly exist as distinct identities; we are that which can be expressed in identities. But what this is, is beyond identity.
Having established this "road" of no identity, we are able to joyfully follow the other "road" of identity; for always we are asked to walk two roads. Finding himself a butterfly, Zhuangzi "fluttered about joyfully, just as a butterfly would." Finding ourselves as human beings (presumably), we live this identity to its fullest. Aware of having no-fixed-identity, we can experience and express joyfully whatever identity it is in which we find ourselves. We are fulfilled in our apparent identity by virtue of the ability to see beyond that identity. Like actors in a play, we give ourselves completely to being our given character, but we do not foolishly attach to it. Yet unlike actors, our non-attachment gives us the power to transform the character we are destined to momentarily play.
You can check out Scott's writings on Zhuangzi here.