A few quotes from the Rabbi on 'Traditions of the Fathers'
in which he clarifies the reason for the law that assigns Jewish identity according to the mother
Children and all living things created in their mother's wombs, and especially those living things that nurse and who grow in their bodily structure from mother's milk, are bound to their mothers from whom they receive, beginning with their conception within the womb's walls and shaping, and in milk that nourishes them, especially humans, in whom speech is an expression of the mind's representations, also received from the mother by her conversation and compassionate caresses, through which she molds and mends his moral character in keeping with her understanding and the education received from her father's house and its surroundings.
Therefore the child follows and respects her, for she persuades him with words, more than he respects his father (Kiddushin 30b). Therefore, were we to determine that a child of mixed [origin] be assigned according to the father, this son would, from the start, be a converted son for his parents, who would waver between two convictions and end up without either, and not only could he not become a person of religion but would be neither humane and moral, for which reason G-d's guidance in this case is to attribute the child to his mother.
Mishpatei Uziel Responsa, Chapter 2, paragraph 6