That she made him a citizen because she was a citizen is not true. The law at the time said she had be past the age of 18 for four years in order to convey citizenship on her own.
She was younger than the limit when she had him, so she couldn’t do it on her own.
That’s only if the birth took place overseas. Not if it took place in the U.S. The governing law was the 1952 Immigration and Nationality Act, Section 301.
That only applies if there was proof the child was born outside of the United States. The mother would have had to be 19 and Ann Dunham was four months shy of her 19th birthday. There is no age requirement for mothers giving birth within the US.
That’s naturalization...