What is the ruling on a brother marrying his sister if there is no blood relation between them, neither from the father's side nor from the mother's side?

This marriage is permissible in Islamic law because there is no legal impediment. There is no prohibitory relation (mahramiyyah) between the brother from the father's side and the sister from the mother's side. Women who are permanently forbidden for marriage are those related by lineage, affinity, or suckling, as Allah Almighty says: "Prohibited to you are your mothers, your daughters, your sisters, your paternal aunts, your maternal aunts, the daughters of your brother, the daughters of your sister, your [foster] mothers who nursed you, your sisters through nursing, your wives' mothers, and your step-daughters under your guardianship [born] of your wives unto whom you have gone in. But if you have not gone in unto them, there is no sin upon you. And [also prohibited are] the wives of your sons who are from your [own] loins, and that you take [in marriage] two sisters simultaneously, except for what has already occurred. Indeed, Allah is ever Forgiving and Merciful." People's disapproval does not affect the legal ruling, because ignorance of a legal ruling does not change its reality. Interaction with them after marriage remains as the interaction of a brother with his brother and sister.