Anchoring ourself into our identity, our story ... means anchoring into our thoughts, our thinking ... it means making our thinking our reality, and thus be like a leaf pushed by the content of our thoughts, also meaning our feelings ... feelings is thinking with bodily manifestation.
So if our thinking about our self "is good", we feel good and happy, ... if our thinking about our self "is bad", then we feel bad and miserable, unhappy. Our existance is completely colored by our thinking, and we are pushed around here and there in the "thinking-wind-direction".
BUT, instead of diving into the content of our thoughts, In ZEN, we leave thinking alone, not stopping it but just letting it come, and go like any sound, taste or smell. ZEN is trancending the conent of our thinking, ... good, bad ... and always has the same silent non-conceptual questions "in the background" What is having this experience? What "I" is left, when there is no thoughts about "I"? What am "I" ? What is having this experience?
So ZEN is dissolving the conceptual mental made "I" and waking up our true nature. Anchor into that, and the wind of our thinking will no longer be able to "push us here and there" as we are anchored deeply in our being, our true nature, right here, right now.