The words by Anne Morrow Lindbergh focus on self-identity resulting from being a stranger to oneself, of being out of touch with oneself.
She claimed that the result of this separation from self is estrangement from others. The clear implication is that the self causes estrangement with others.
However, according to the root definition of “estranged,” the primary source of estrangement is not the self but the family of origin.
Consider the etymology of estranged from OxfordDictionaries.com
Late 15th century: from Old French estranger, from Latin extraneare ‘treat as a stranger’, from extraneus ‘not belonging to the family’, used as a noun to mean ‘stranger’. Compare with strange.
Lindbergh's statement might accurately describe adults who experience self-doubt. However, the root meaning of the word "estrangement" is about being treated as a stranger who doesn't belong in the family.