The best way to understand Evangelion's uniqueness is to become familiar with all the tropes in anime/manga. Then it becomes obvious how Evangelion turned them all on their heads in some of the darkest possible ways -- and became one of the biggest hits of all time in Japan; something truly "generational" like Star Wars.
Anime and manga had long excelled at high drama; the space operas and melodramas and such.
And the creators of anime and manga were certainly not lacking in self awareness; there were lots of parody and comedy in anime and manga over the years, much of the light parody focusing on other works in their own genres or the general tropes found in those works.
But Evangelion was arguably the first major work in which those worlds collided. It never breaks the fourth wall, but the subversion of anime's tropes and conventions is utterly central to its drama.
1. The protagonist is a teenage boy. No surprise there -- the protagonist of any show is usually representative of the target audience. But instead of a simple "boy becomes a man" narrative, he more or less descends into depression and PTSD. The act of piloting a robot in bloody combat against incomprehensible is not healthy for him (because it wouldn't be, for anybody) and he is simultaneously loathable and also incredibly sympathetic because god damn is he put through hell. He responds to these traumatic events in ways that we might - imagine if your abusive father abandoned you and then forced you to become a child soldier? How screwed up would you be? He's probably the first anime protagonist to say: "Wait, this isn't fun. This is seriously fucked up." (And going further down the rabbit-hole of self-awareness, Shinji is painfully aware of how messed up he is)
2. Giant robots. They are a staple of anime for a number of reasons, chiefly because they sell toys, and toy sales help to finance these shows. Evangelion give you "robots," alright, but they are utter nightmare fuel in ways both visual and psychological.
2a. A trope in anime is when the pilots of giant robots "feel" their robots' pain. Evangelion takes this to the darkest and most literal extreme possible, with damage to the robots resulting in physical and emotional damage to the pilots.
3. Weirdest. Love triangle. Ever? It would be spoilers to say more, but it makes Game of Thrones look wholesome by comparison. Romantic tension is a given in... just about anything... but maaaaaaan does Eva make it weird. Another expectation, subverted, darkly.
4. Misato is something of a surrogate mother to Shinji. That gets, uh, interesting.
That said, Evangelion is not good just because "it's dark." There are lighthearted moments, and happiness, and some characters grow and become better people. It never feels "dark for the sake of being dark" in the way that some series like Game of Thrones are, where you almost dread the happy moments since those happy moments because they are immediately followed by some sort of gruesome tragedy, as if the viewer themselves is being punished for daring to be happy.