Interesting story and interesting topic!
Although I have heard similar things from others as well as you describe about Neon Genesis Evangelion, I do not know why, but it didn't have as much of an impact on me as on many others. I still think the symbolism of the show is uniquely interesting in quite a few ways and it captures inner mental turmoil in an intriguing way, but it did not resonate as much with me personally. This might be because I only got to watch it a year ago, i.e. after I have had my fair share of experience when it comes to psychology and especially philosophy, and am already old
The question is quite difficult though. I can't say that an anime has had a sudden/discrete impact on me compared to unexpected events such as the death of a friend. But at the same time, I think there are many anime that have had an effect over time. Something that got me to think about something in a different way which later led to more substantive changes. This does happen quite a bit since I tend to over-analyze anime or at least see this as an example to a more general philosophical question which on the face of it is not related to it.
To give you an example. I watched a part of Attack on Titan (I didn't continue it since it was too much gore for my taste). When I saw the whole Titans eating humans thing, I did start to think about vegetarianism and cannibalism. Now, vegetarianism is something I had thought about before quite a bit because of a colleague who has worked on it, but cannibalism not so much. So when I now think about cannibalism (which granted does not happen too often unless I think/read about lab-grown meat), Attack on Titan might be one example for it.
Apart from that I would say that all of my favorite anime resonate with me in one way or another and have influenced the way I see or think about things for years.