So in their day, the Nazis were allied to Japan, a non-white empire. Hitler commended Jessie Owens, while it was white Americans who snubbed him. And of course, the people the Nazis hated the most were just other white people like Jews and Russians.
At the same time, white supremacists in the US terrorized blacks and ran Jim Crow oppression across the south.
However, today we view these groups as one and the same.
How did this happen?
I don't think the premise of your question is true. Who views them as the same? "Nazi" refers to a specific historical group and political ideology. And "white supremacist" is more of a blanket term referring to many different types of groups and ideologies.