With the immunization laws in 1961, Obamas mother may not have been able to get her jug eared clown out of Canada if she had him listed as a British subject born in Canada, which is why they may have needed a phony BC showing he was born in the US.
I remember as a kid in the early 1960’s the whole town had to line up for immunization a couple of times.
People simply weren’t allowed to come and go into and out of the US without the right shots.
If you were born in a foreign country it would be very natural for your BC and vaccination records to be together.