The question regarding the address used still stands.
IIRC not uncommon to use the place of birth as it does not vary. Place of residence does. It’s a Genealogy thing.
The HI DOH just acted on the info given them. The address was given, not verified. Only took one person to file the initiating paperwork (fraudulently at that) to set this whole fiasco in motion.
One viable theory that BHO & Mama weren’t even in HI; she informed a relative of the birth, and that person went and filed the fraudulent forms.