I think personality does not change. It evolves. Cut from the same cloth, but colors are changing with the seasons. They have to, otherwise it would be hard to adapt to new situations and you'd be causing yourself distress. So in a way, there is no stable sense of identity.
However, I think there are fundamental aspects of oneself that will never change. I feel that sometimes our "story" in this life and our perspectives revolve around those aspects that are core to who we are. Other aspects are adaptive and necessarily so to increase our chances of survival in this world. Maybe the former drives the evolution of the latter. We grow up, we mature, yet we still distantly feel the child that we once were.
All this is to say that if by "stable sense of identity" you mean carrying out the same personality and exhibiting the exact same behaviors, then I don't think this is possible, and if it was, it would not be healthy. But if you mean know who you are and what your essence as a person is, then yes, it is possible, but I think it comes with age and experience. And that would definitely require time for it happen, like you said.
To answer the question of what it feels like; having a stable sense of identity inspires confidence and strength when life is treating you well. And it inspires hope and faith in yourself when things get difficult and doubt fills up your thoughts. Becoming familiar with who you are as a person and what makes you tick and how you operate can definitely provide a sense of security and trust within, especially when your only company happens to be yourself.
On a related note, Identity can not be described with words, I don't think. No matter how much we try, there isn't enough vocabulary to describe such complexity. I would say Identity is an intimate feeling about one's self. A kind of awareness that is developed only through life experience. If one ever wishes to gain insight on what their identity could possibly be, I would say it is that one "common theme" that is present in every chapter of a their story.