Going nazi punching is at this point synonymous for going on a witch hunt. Nazi punching represents a villainization and persecution of a group of people based on their political beliefs. This can not be allowed in a fair and equal society. People should not be subject to different standards based on their political beliefs. In fact, political ideology is considered a protected class under some laws and treating someone differently based on their political ideology is an illegal form of discrimination. If someone can prove you didn't hire them or didn't serve them because you disagree with their ideology, they can sue you for wrongful discrimination and it would be justified.
Part of the problem with nazi punching is that violent people were going on witch hunts for nazis. The other problem is stupid people were using the presumption of nazi status to justify mistreatment of other people whether they were actually nazis or not. This mistreatment included physical violence, threats, bullying, and ostracization. Nazi punching is based on hysteria, fear mongering, hyper vigilance, and virtue signaling. Another problem with nazi punching is that there just aren't that many nazis around and they aren't doing anything to you. Another problem, nazi punching is a symptom of a breakdown of our freedom of speech. You can't believe in free speech if you don't respect the speech of your enemies. Good ideas beat bad ideas, thugs beat nazis.
If you know what a nazi is, you would be able to appreciate that it stands for national socialism, and yes the third reich was classist and racist. To call someone a nazi implies antisemitism and white supremacy, and should be seen as slander and defamation of character. The term is so loose now, people seem to think it merely means a particularly ugly fascist. You'd have to have something decent to show that someone calling you a nazi meant to call you a racist. If you think about it like that though, it is a ball-less slander because it is so ambiguous. You'll have to show that it did damage to sue.