Mombasa wasn’t under Zanzibar’s sovereignty, they were under Kenya’s as the British government paid for the lease (or tribute as they called) it of the 10 mile strip including Mombasa for nearly 75 years before Zanzibar ceded the 10 mile strip. The civil records would have been under Kenya since Kenya had jurisdiction over the area they LEASED.
It’s like if you rent an apartment or house and have to list your address, you would list the place you reside and all activity would be on you. Not the landlord.
It would be better to say that Mombasa, while leased to the British, was nominally under the Sultan of Zanzibar’s sovereignty but was actually under British government, authority, control etc.
Sovereignty is the broader, more abstract concept. By “leasing” the coastal strip, the Sultan retained, in name only, nominally, the sovereignty (he had to have it else he could not “lease” operative control to someone else). But he leased, ceded, control, de facto, on the ground administration, to the British.
So in terms of people’s perception, Mombasa was “part of Kenya.” On paper, nominally (and I’m sure people were aware of this, but at a secondary level of awareness), it was still under the Sultan’s sovereignty.