Under current Oklahoma adoption law, a judge will make that determination. But the judge will base that determination on facts (evidence that is either document-based, i.e. an exhibit, or testimony-based, i.e., a witness, or some combination of them both). And, there's a framework in place on how the law applies to those facts.
In short, there are two tests for making that determination. The first test is whether the child is an enrolled member of a federally-recognized Indian tribe. The CHILD--not the child's grandparent, for example. Also, ENROLLED--not merely eligible for enrollment. Also, a FEDERALLY-RECOGNIZED Indian tribe--there are many that are not, but well over 500 that are. That is one of the two tests the judge will use. If all those parts are present, then the child passes the test and the federal ICWA will apply to the adoption case. But if the child fails that test, the judge will look to see if the child can pass the second test.
The second test has two prongs, (i) that the CHILD is ELIGIBLE FOR ENROLLMENT in a FEDERALLY-RECOGNIZED INDIAN TRIBE and (ii) that the child's parent is an enrolled member in a federally-recognized Indian tribe. On the second prong, watch out for the word "parent." It has a very specific definition and it might surprise you who is not covered by the definition.
Why would a prospective adoptive parent of this child care about the determination? Because the federal and, if the adoption concerns a child born in Oklahoma, the Oklahoma Indian Child Welfare Acts will apply to the adopton case filed in an Oklahoma District Court. (Interestingly, the federal and Oklahoma ICWA do not address themselves to tribal courts, so those courts are unlikely to concern themselves with ICWA provisions).
Why would the child care about the Indian child determination? Because that child is a precious resource to that child's tribe (or tribes), and the tribe (or tribes) care mightily and have much to offer that child--on levels too numerous to list in this FAQ response.
And as for the third member of the typical adoption triad, the child's parents (there's that word again), failing to follow the federal and Oklahoma ICWA when there is a duty to do so means that their adoption parenting plan is legally VULNERABLE/UNSTABLE/RISKY. Following the provisions when they are required provides many extra protections for parents, while failing to follow them shortchanges the Indian child's parents (one of whom may not even have Indian blood but is still entitled to these extra protections as a parent of an "Indian child").
Also, a tribe's interests are deep and wide; they run strongly throughout the adoption case when an Indian child is involved. That is why many adoption practitioners and other adoption professionals consider the tribe a necessary party--or at least an integral part--of any adoption of a child who has or may have Indian blood, even before a judicial determination is made, and often stay in communication with the tribe even if the child meets only one of the two prongs in the second test or otherwise falls outside the definition of "Indian child" under the federal Indian Child Welfare Act.