We have a couple of black sapote trees in our yard; sounds similar.
Did you know… black sapotes are of the same family of persimmons? If you focus, you’ll notice that they taste very similar.
Is an american cousin, Persimmons are Diospyros kaki and Black sapotes are Diospyros nigra, so technically is a black persimmon. In US there are other obscure native persimmons also like Diospyros texana