To tell you the truth, the “Nazi” metaphor was always too much for me. It treated lightly something I wasn’t ready to treat lightly. I am a veteran of a good deal of post-WW II anti-Nazi propaganda. I remember Nazi propaganda from a time when the Nazis were still winning. And, of course, I have seen many anti-Nazi movies. So treating it only metaphorically still doesn’t feel quite right, and there are several reasons for that.
First, you may not have noticed, but real honest-to-goodness Nazis are coming back. These are the classic Nazis with the racial paranoia and the lust for authority and the attraction to violence. The guy who edits a Nazi newsletter is a grammar Nazi and it isn’t even a joke. The idea of Rudolf Hess (no relation) as a grammar Nazi actually is a joke and I had not heard it until today.
So even if it was funny back in the Cold War days, it isn’t funny anymore and may not even be safe anymore.
Second, the emphasis on grammar obscures a much more pervasive problem with language generally. Consider this example from Michelle Obama.
“I remember there were kids around my [Chicago] neighborhood who would say, ‘Ooh, you talk funny. You talk like a white girl.’ I heard that growing up my whole life. I was like, ‘I don’t even know what that means but I am still getting my A.’”
The problem here is racial identity. The “kids around her neighborhood” had a model of language use in mind that specifies how “one” should talk if she is to be considered “one of us.” The threat is that if you don’t make the same mistakes we make, we will reject you and may well persecute you.
But it is class identity, too. In South Chicago, the two are intertwined, but in the Middletown, Ohio of J. D. Vance’s Hillbilly Elegy, they are not. The demand that you “talk like us”  has, in Vance’s book, to do with class norms, not race norms. If you use speech that sounds like the kind the kids of the professional classes are taught to use, you will get the same kind of threats Michelle Obama got.
The essential part of the “Nazi” metaphor is the enforcement. It is not any part of the joke that Nazis have better grammar than “normal people.” It is the intrusiveness of the enforcement that suggests Nazism. And in the Obama example, the intrusiveness belongs to the users of incorrect grammar. The demand is, “You will use the grammar we use or you will pay the price.”
Does that sound Nazi-ish to you? It does to me.
My father used to tell a story that had to do with language and character. I am going to tell you the story and then reconsider it from the Language Nazi standpoint. Dad used to say that a boy  who says “My old man ain’t got no right to bitch like he’s doin” and one who says, “My father has no right to complain the way he is” are saying the same thing about the father. They are saying different things about themselves.
The story itself implies that it is the character (read “social class”) of the boy that controls the form of the speech. In Dad’s use of it, it also implied that the first boy’s expression (=character) was inferior to the second boy’s. Dad could easily have become a Language Nazi by cruising the neighborhood and correcting the errant expressions of neighbor boys. He didn’t do that.
But the neighborhood boys could also establish the “my old man…” formula as morally superior and punish deviant uses. “Deviant” in this case would point particularly to the “My father has no right…” formulation. And if they enforced that usage, excluding and punishing the people who said it “the wrong way,” they would be Language Nazis in the same way, but in the reverse direction.
The two seem remarkably similar to me.
You might respond that there is an objectionable “snottiness” about the traditional Grammar Nazi kind correction, a common instance of which is the confusion of “your” and “you’re.” (As the example shows.) But I would argue that there is also a remarkably pervasive snottiness about language that is “too white” or “too middle class” for the present group of hearers.
“Political correctness” is a good contemporary example. The term itself is a protest by conservatives who are tired of being “corrected.” “Correct” is a notoriously weak norm.  If, for instance, the indigenous peoples of North America  find being called “redskins” or “Indians” to be offensive and if we have no reason to want to offend them there is no reason we couldn’t call them what they want to be called. But the criterion that marks such a choice is that it is collegial or civil or even generous. Justifying it on the grounds that it is “correct” is more like a criticism than like a justification. And, in fact, that is how it is used.
But if calling these people Indigenous Peoples is going to bring you ridicule and exclusion from one group and calling them “redskins” is going to bring you ridicule and exclusion from the other group, which are the Nazis? Does it make any sense anymore to label one of the groups that patrols expressions and punishes non-conformers “Nazis” and not the other group?
And that brings us back to the Grammar Nazis with whom we began. So long as the upper classes use a grammar of a certain sort, there is going to be a temptation to call it “standard” and to use it to measure the speech of others, who are presumed to aspire to upper class status. If the speech of these people is no longer widely admired–and that is one of the likely effects of the recent failure of upward mobility, then “correct” will have a specific social location, say a street gang, or a specific context, say a job interview. 
The people who punish you for using formulations they don’t like will have earned the Nazi label without any help from you.
 Pause for one delicious moment to substitute “Talk as we do” and then close that part of your mind and move on.
 My father had four boys to raise, so the stories were always pitched in the direction of male illustrations.
 “Indigenous” would have to mean the people who were living there when we became aware of them. It could not mean the people who were there before and whom the people we call “indigenous peoples” displaced. “Indigenous” is a slippery sort of notion when you look at it.
 It is, in that way, like the “conventional” in John Kenneth Galbraith’s widely used, “conventional wisdom.” It is as if the assembled agreement of all scholars made a point too dull to be really worth making.
 Bergen Evans, co-author, with Cornelia Evans, of one of my favorite dictionaries A Dictionary of Contemporary American Usage, used to say that he didn’t correct the speech of anyone who didn’t pay him to correct his speech. Evans is the ultimate non-Nazi.