No problem. This purported document is from 1964. It is a Kenyan document dealing with an event that supposedly took place in Mombasa not quite three years earlier. Mombasa was in the Coast Province of Kenya in 1964, so the official in charge of the birth records for Coast Province was responsible for checking the records of births in Mombasa; presumably whatever British Colonial unit was responsible for Mombasa when it was part of Zanzibar transferred its Mombasa records to the Coast Province of Kenya when Mombasa became part of Kenya.
If this document was purported to be a 1961 document, then we’d have a problem.
The document may be a fake. But Mombasa-Zanzibar-Kenya business need not be a deal-breaker. Someone would have to trace the exact structure of colonial and early independence bureaucracy in Zanzibar and Kenya to be sure. There must be records somewhere of which units were responsible for what when.
Mombasa came under the control of the sultan of Zanzibar in 1840 and was leased to the British in 1895. From 1905 to 1907 it was the capital of the British East Africa Protectorate. In 1920 it became capital of the British Protectorate of Kenya.