Mombasa was a technical territory of Zanzibar, but the 10 mile stretch of Indian Ocean coastline (Including Mombasa) was and had been for 74 years, leased to Kenya by Britain.
The record keepers of Mombasa would be keeping records for KENYA, not Zanzibar.
About Obama’s Kenyan grandma who claims to have been present at his birth—how far did she have to travel from her (presumed)village to that hospital? Maybe she was asked to be present since no other female relatives of the unborn baby could be there, particularly if the young Ann was sort of temporarily “trapped” in a foreign country with no other connections.