My brother and I grew up in the 60’s and 70’s. There were a lot of WWII movies on TV in those days, and he studied up on the Nazis, so I’m sure he would be familiar with them. Random fact: Hugo Boss designed the SS uniforms.
Yeah, but I'm sure your brother isn't from about 320 years after the SS last wore their uniforms, and as a bonus, I doubt he's from another planet. And even if he were from an Earth-centered military organization from that future time period, I wouldn't expect him - much less both him and the closest-ranking officer in his unit, one of them an alien - to have studied Earth history to such an extent that they could identify rank insignia from Earth's comparatively ancient military forces with precision. (Particularly forces whose aesthetic, insignia, and even ranks themselves had only very tenuous connections to other historical military units, and familiarity with which would not really have any conceivable application to the officers' work duties - insert use-of-Paramount-costumes-as-basis-for-episode-joke here!) Now, if Kirk and Spock had been U.S. military officers in the late 1960s, maybe they would know SS (and Gestapo) ranks by sight. Maybe.
There wasn't really any convenient way around this in the script, but I did find it interesting.
As for Hugo Boss, he wasn't a designer. His company apparently (and sadly) may well have produced some of the SS uniforms, though.