Somewhat related to a recent question, where in answers we find out out that several AfD members won defamation lawsuits for being called "Nazi", Höcke was called "new Hitler" and not in jest (satire being the caused one "Nazi" defamation case was dismissed, when others were not). Did Höcke sue over this and did he win? (I see in another article that he apparently threatened one such lawsuit, but I'm unsure if he went through with it.) Likewise, he was called "AfD fascist". As far as I can tell these exact epithets originated in the British press, but at least the former was repeated in the German one. (I'm not sure how the German law deals with quotes of defamatory material. In Common Law, there's not much difference.)
You are allowed to call Höcke a fascist, but you are not allowed to say that a court said that Höcke is a fascist.
I am not aware that Höcke himself sued anybody over being called a fascist (or that he has been called "The New Hitler" in Germany - a blog by concerned citizens, which you link in your question, is not representative of the German press, and in fact they say that only foreign press dares to say what they allege to be "the truth").
In 2019, the AfD held a little rally in Meiningen. A group of citizens announced that would have a counter protest with the slogan "Protest Against the Racist AfD, Especially Against the Fascist Höcke". The municipality of Eisenach (which Meinigen is a part of) wanted to ban the term "fascist".
This ended up in the "Verwaltungsgericht", which my dictionary insists means "district court", but which is actually the administrative court that adjudicates conflicts between citizens (here, the protesters) and the state (here represented by the municipality of Eisenach). The freedom to have an opinion and disseminate it via all legal channels is part of the German constitution, so trying to ban this was a pretty bold move by the city of Eisenach (libel is still illegal, but the bar is very high).
The court pointed out that they did not view "Höcke is a fascist" as a statement of fact, but as a moral judgement, and would therefore only examine the question if the moral judgement is legitimate. They concluded that indeed it is, based amongst other things on the fact that Höcke had in written form embraced tenets of national socialist ideology (such as the idea that Germany needs a "Führer", and that an international conspiracy aims to replace the German population).
Again, Höcke was not part of the proceedings, and the article linked above claims that he never (at least up to the date of the article) sought legal action against people calling him a fascist (he is probably aware, given the stuff he says in public, that his chances would be essentially nil).
did successfully sue succeeded in getting an injunction against liberal politican Sebastian Czaja (MP in the Berlin state parliament) for the claim that the court had called him a fascist.
The difference between "the court allowed to call him a fascist" and "the court called him a fascist" might sound specious, but the point is actually not very subtle. The whole raison d'etre of the Verwaltungsgericht is to protect citizens against state interference, so they would never be allowed to make moral judgements on citizens themselves (also presumably saying "the court called him a fascist" might confuse people into thinking that he had committed a crime, which is nothing that would be determined by an administrative court).