That may be true for child support payments. It is not true, however, for citizenship claims. There must be a blood relationship in order to claim U.S. citizenship. See the U.S. Department of State Foreign Affairs manual for details.
The claim is not verified by the U.S. as to their being an actual DNA relationship.
A DNA relationship is not necessary for one to be the “legal father”.
If Barrack 0bama Sr. is listed on 0bama’s BC; he is 0bama’s “legal father”.
I really doubt that they do DNA to determine parentage.
USCIS, not the referenced manual determines eligibililty. The USCIS allows the naturalization of adopted children. Also, your referenced manual, for what it matters, states that an adopted child is defined as a child of the parent.