Memory Passing, Reincarnation
When the male decides to mate with a woman, males may inject parts of their unique memories into their spirit, and pass this piece of spirit into the woman. They can pick which memories to share, and how much of their spirit to give.
By mating with the same woman multiple times, the male is given a chance to inject more and more memories and characteristics of himself into the creation of the child. This can include his experiences, skills, and even, deadly secrets.
Theoretically speaking, if the male exhausts almost all of his spirit in the creation of the child with the woman, the child can be seen as a resurrection; he will inherit the genes of the woman, but inherit the memories of the father.
Patri-lineage of Skills and Memories
Patrilineality is a common kinship system in which an individual's family membership derives from and is recorded through their father's lineage. This is normally how position and inheritance is decided in history. It generally involves the inheritance of property, rights, names or titles by persons related through male kin.
In this case, Patrilineage would mostly consist of spirit (memories), and name. Due to the ability for males alone to share their spirit down their lines, male lines would be capable of propagating memories as an inheritance and property.
Patrilineality in Spirit, The Ritual of Return
You could say that the inheritance of the spirit itself is representative of the patrilineality. By inheriting the memories of the fathers, the child is able to gain an advantage over others. If their father was especially gifted with certain jobs, or talents, they could pass these skills along to the child, although they would lose it themselves. The father could leave a part of his most useful skills, to entice the child, as well. They could also pass down a portion of the memories that were given to them by their father, grandfather, and so on.
Something that can happen to guarantee a degree of Patrilineality would be the succession of the child from the father. In cases where the child is created from the spirits of many fathers, the child's life will be dictated and influenced by the spirits and memories of its fathers.
Not only that, in the case of children with powerful Matrilineal lines, the women have more choice over what men to mate with, and the men would wish to have the children return to them and recognize them as the father.
Thus, there would definitely arise a ritual of some sort, let us term as the 'Ritual of Return'. The Ritual content would be the child, formally recognizing his father, who will accept the child into his family, and pass down the complete part of his own formal inheritance, if he had given only part of it before. The child, would then accept the father's name as part of their own name. As a gesture of the ties between them, the mother's family would also pass down a sort of child-dowry to the father, and accept the father into the family.
Fathers who have no property, would naturally wish to gain the recognition of a powerful Matrilineal line, who owns physical properties. The competition between the fathers in donating more Spirit and precious memories, to compete for the acceptance of a Matrilineal line, would thus be very fierce. By encouraging this competition, the most powerful matrilineal families would ensure the quality of their children, with heavily abundant spirit and the most precious memories.
To summarize, the role that patrilineal lines play in a child's life is the inheritance and influence of various spirits, memories, and skills from their father(s). This stays relevant because the matrilineal lines would encourage such inheritances to guarantee the quality of their children in spirit and skill.
These skills and memories would influence a child's life forever, and in the case of the males, extremely powerful skills, talents, and memories can be passed down generation to generation due to the spiritual nature of mating.