Yes they do it's actually a big part of American culture to identify as your ethnicity domestically. They just considered themselves Americans first internationally
I see your point but this is because to the rest of the world he's not Russian, he's American. But if he was talking to Americans he and they would say he's Russian
In that case he is both.
He spent a lot of his life in russia and speaks the language natively, but migrated to the US a long time ago so is also an american.