As per your query it is understood that you need information regarding the inheritance of the adopted child without any documents of such adoption.
An adoption without valid paper work i.e an adoption which is not admissible under the Hindu Adoptions and Maintenance Act, 1956. But in the absence of any documentary evidence to prove that he have been adopted by your father in law the burden of proof lies on him and through oral and other circumstantial evidences he can prove his case i.e if he is really adopted by your father in law.
In the famous case titled Addagada Raghavamma And Anr vs Addagada Chenchamma And Anr
Equivalent citations: 1964 AIR 136, 1964 SCR (2) 933 the court held that "It is well settled that a person who seeks to displace the natural succession to property by alleging an adoption must discharge the burden that lies upon him by proof of the factum of adoption and its validity. Their Lordships went on and said that 'ordinarily' an only son is neither given nor taken in adoption."
"An adoption would divert the normal and natural course of succession. Therefore, the court has to be extremely alert and vigilant to guard against being ensnared by schemers who indulge in unscrupulous practices out of their lust for property. If there are any suspicious circumstances, just as propounded of the will is obliged to dispel the cloud of suspicion, the burden is on one who claims to have been adopted to dispel the same beyond reasonable doubt. In the case of an adoption which is claimed on the basis of oral evidence and is not supported by a registered document or any other evidence of a clinching nature, if there exists suspicious circumstances, the same must be explained to the satisfaction of the conscience of the court by the party contending that there was such an adoption."
If he is able to prove in the Court of law that he was adopted by your father in law then he has the right in his property and he can claim it. This situation arises in circumstance when the will is not made by the deceased person.
As to be the legal heir he has to obtain the legal heir certificate and for such certificate he has to prove that he is the adopted child but without the required paperwork just by oral evidences and by the name in ration card he’s case is really weak.
To know the procedure to delete his name please refer to the below link.
Hope it helps!