..it is healthy and necessary to associate *a piece* of our identity with our work.
For the entire history of our species, we have attached meaning to our role within our group. Hunter. Gatherer. Cook. Child protector. Warrior.
These roles have meaning because they signify our contributions. They represent how we spend our time and how we create value for one another. We would not survive as a species without playing a role in creating progress and opportunity for one another. Our roles represent the reciprocity principle.
Saying that we should not attach identity to our work is merely a way of protecting ourselves from the psychological impact of being stripped of that aspect of our identity. It is painful to quit a job precisely because our identity is tied to our role in the group. It is even more painful to be fired for the exact same reason. I have done both and I cannot honestly say that one is easier than the other.