I would assume that the US probably did have an Embassy in Kenya at the time. But, to the best of my knowledge, there was never any American law requiring American citizens to register at a US embassy, except maybe in hostile nations or in friendly nations where the American citizen voluntarily sought services that the embassy could provide. Kenya, under British rule, would most definitely have been considered friendly.
I certainly think that if Obama's American mother was in Kenya at the time of his birth, she would not be required to report his birth to the American embassy unless she, e.g., wished to begin the process of applying for American citizenship for him. If he were born in Kenya in 1961, he would not be a natural born citizen, but rather would have to be naturalized if his mother were seeking American citizenship for him.