Last time I checked, Jamaicans were black (well, the vast majority of them, I suppose there are a few Euro-Jamaicans?) And I'm sure Ms Harris is an American citizen, born in the country. Ergo she's a black American. Though also Indian-American.
She's almost certainly got slave ancestors. Just not ones who were exploited on US territory.
There are surely some differences in the historical experience of those who came via the independent nations of the Caribbean. But so what? He's clearly being a xenophobic US nativist, by denying her American identity.
I don't believe the guy is Russian . According to this article
https://www.huffingtonpost.co.uk/entry/kamala-harris-bot-twitter-birther_n_5d1692d1e4b07f6ca57d4fd8
If that's true, then it's a bit ironic he changed his name to the rather Russian-sounding "Alexander" (after the Czar?).