As best as I can understand, some anonymous person in the early modern period, 1600s-1700s made up this idea that the ancient Romans did this funny salute. It’s an urban myth. There’s no evidence that real ancient Romans ever used this practice.

Anyhow, the idea floated around for a while; it shows up in paintings in the 1700s. It was picked up by several political movements in the early 20th century, including Bellamy in the United States, and Mussolini, who was a big Romaboo. The Nazis did get the idea from their Italian allies.

The German Nazis took the idea from their allies, the Italian fascists.

The United States was doing a similar thing with the Bellamy salute from the 1890s up until they weren’t buds with the Nazis any more.

There is art work going back to the 1700s depicting ancient Romans using the salute. Meanwhile, the idea has staying power: I recall that this salute appears in the 2005 HBO series Rome.

So it’s an urban legend among classicists that got picked up by several politicians in the early 20th century, and notably the Italian and German fascist movements.