I think the reference is to the Brown Shirts and Black Shirts in the period before the takeover was completed.

There is a degree of conflation, perspective from history.

The party uniform of the NSDAP was brown.

Even after they come into power.

As the party colours of the right-extremist/fascist parties in Austria are blue, they are sometimes called "blaun" an artificial combination of "blau" (blue) and "braun" (brown), to "subtly" point out their fascist tendencies.

The origin of the brown shirts worn by the SA was an accident of history. Not long after the end of WWI, some uniform maker got stuck wirh a huge run of unsold brown uniform shirts. The early Nazi Party (NSDAP) scooped them up on the cheap, and adopted them as part of the SA's uniform.

The SA were full of shit wirh the Stormtroopers name BTW: the name originaly referred to elite "push-men," shock troops kept out of battle until it was time to attempt to storm an opposing trench under a rain of artillery as suppressive fire. Most WWI armies had them under various names, the Italian Arditi were probably the most capable. Germany's stormtroopers managed more than once to force a breach in the stalemated front, only to find the regular army could not exploit the breakthroughs.

This was due to reliance on horse-drawn transport, and meant the war continued until all sides were utterly exhausted. Germany's WWI stormtroopers had a casualty rate of about 25%, not sure if that was total or per battle but the latter seems lilely. On the front they were often treated as rock stars. Imagine Manowar or Sabaton on a front line in Ukraine, that's the kind of attention they got.

Years later the Nazis appropriated the name for street-fighting hooigans who never had to storm anything tougher than a beer hall. Beer halls defended by Communists could put up a good fight, and were nearly a match for the Naziss but not quite. The Beer Hall Putsch proved these so-called "stormtroopers" could not even handle armed police much less fight a serious military unit. A tough antifa-held beer hall was their absolute limit.

Yet when you say "stormtrooper" today people think either of the SA or the white-armored regular troops of the Empire in Star Wars. The "push-men" of all the WWI armies who tried ro break the stalemate and thus stop the meat grinder are all but forgotten.

The second and third meanings both fit ICE, they'd shit their pants trying to take a trench defended by an army wirh cannon and machine guns, say in Ukraine today.

Today's version of the SA would be the Proud Boys, Oath Keepers, and 3%'ers. Next to even regular troops they look like out of shape reservists or maybe just cosplayers. I've faced them in battle myself several times. They are best compared to fake US Navy SEALs wirh store-bought tridents.

And Mussolini and Mosley's lot, the latter satirised by P G Wodehouse as Spode's Black Shorts.
An unrelievedly vile collection.

Yes! Now that Trump pardoned the J6 goons and wants to pay them, I'm looking at Oath Keepers and Proud Boys in a Freikorps perspective. Or Wagner group.

I think of those outfits more as Trump's SA (so-called stormtroopers). They are at risk of being replaced by ICE just as the SS replaced the SA.

For what happened next, look at Nazi Germany's "night of the long knives."