It is other income, not salary
The key point is the category. Your own pension, from your past employment, is taxed as salary. But a family pension, one paid to you because the person who earned it has died, is not your salary, you never worked for it, so it is taxed as income from other sources under Section 57(iia). That distinction decides the deduction you get.
Because it is not salary, it does not get the salary standard deduction of ₹50,000 or ₹75,000. Instead it gets its own, smaller deduction: the lower of ₹15,000 or one-third of the family pension, and under the default new tax regime that ₹15,000 figure is raised to ₹25,000. So a family pension is taxed on its amount after that modest deduction. Confusing it with the salary standard deduction, and claiming the larger figure, is the common mistake, and it does not hold up on assessment.
When it is fully exempt, and the NRI position
Some family pensions are not taxed at all. A family pension paid to the widow, children or nominated heirs of a member of the armed forces, including the para-military forces, whose death occurred in the course of operational duty is fully exempt under Section 10(19). And a family pension received by the family of a gallantry-award winner, such as a Param Vir Chakra or Vir Chakra recipient, is fully exempt under Section 10(18). So if the pension arises from one of these, there is no Indian tax on it at all, and it is worth checking whether yours qualifies.
For an ordinary family pension, an NRI is taxed in India, because it is Indian-source income, on the amount after the small deduction. An Indian payer may or may not deduct tax at source on it, so it needs to be reported and reconciled on a return. And as with other pensions, a tax treaty may affect where it is finally taxed, so the treaty position is worth checking. A practising CA classifies the family pension correctly as other income, claims the right deduction, applies any full exemption, and reconciles the Indian tax.