In the early 1930's Antifaschistische Aktion (abbreviated to antifa) was founded by the Communist Party in Germany to engage in direct action and street fights with the rising Nazi party and facism.
It has continued as such and though the far left types don't like to admit it, or internet pseudos don't understand, it is the nature of the beast. The far left do want to smash the current world order (capitalist system) meaning that far left and anarchism can be discussed interchangeably by the media etc.
The different movements have the same name but different things though.
The original organisation you mention was the paramilitary wing of the KPD. In all honesty, the 1930's was a very particular time, as was Weimar Germany. Democracy was quite modern and not as cemented as it is, neither were reforming organisations. Anarchists were certainly not involved in that section of it.
The AFA that followed in the 70's, 80's was a bit a mix between builders, dockers, some trotskyists, anarchists and some hunt sabb guys. It looked very different to the AFA you have mentioned.
Those people who use ANTIFA now are a different organisation again. I don't rate them much if I'm honest. Some crossover with some lefty anarchists. I'm not sure how many structural communists really get involved with them now. They're a bit of an embarassment.