So the bar for doxing is definitely a low one, but in this particular case it isn't met. I can see why Musk is irritated that that account exists, even more so because it didn't go away at the first request by someone as powerful as him, and that makes it personal. See the whole saga with that diver for a typical response. But that doesn't mean that the person manning that account is doing something illegal and that is the bar which Elon Musk himself set not all that long ago, and which is what makes this news.
If he had been a bit smarter about this he would have just said: "I'm irritated by you, this is my site and you're gone". That would be that. But now there are all these logical pretzels why this is illegal and all that other stuff that people - and Musk - do on twitter is not because 'free speech'. The two are incompatible, and he knows it.
I know people wish this weren't the case, but your address isn't exactly private information. Anyone can find it easily for anyone else.