Actually one didn’t be in Kenya physically to have a birth recorded, all they would have had to do is visit the nearest British consulate for the appropriate paperwork, fill it out and send it in. (Remember when Obama was born Kenya was actually a crown colony, not attaining status as an independent country until 1963). Canadians living in the U.S. apply for Canadian citizenship for their U.S children all the time.
Recording births isn’t unusual. But, word is, there is a birth certificate. That is different. The records would also confirm ethnic origin. That is also significant.