It's defo largely subjective, but there are a couple of shows that are undoubtedly iconic and profound to a bunch of people....NGE and Death Note come to mind. To me profound would be that it changes how you think long-term, makes you feel a lot of emotion (other than just excitement) or gives you that feeling of what might be mental clarity, like you just feel different and the world around feels different - not sure how to explain it, but if you've felt it you probably know what I mean....like the feeling after a funeral (especially the graveyard. In Muslim and I believe Jewish and Jamaican funerals (at least my uncle's friend's mum`s funeral which he filmed for him and I saw a bit of) the family and other funeral goers fill in the grave, so you're there a while. Dunno about how others do it. Usually the family cleans the body too) or maybe you can get there through meditating in silence or with certain types of music.
What are some shows you mean from back in the day, Nila?
Post-2020 I don't know any anime really. The last one I watched that I thought was beautiful was the movie Your Name, from 2016. Before that it was the Inuyashiki anime, from the MC just being uniquely inspiring in high-stakes moments (bit of a mary sue I guess). Blue Period seems to well-regarded and came out very recently. Beastars could be considered profound by its fans, since it covers social issues in a slightly unique way, but I wouldn't call it profound myself.
I'll rec Mob Psycho, cuz why not. It's a nice coming of age story about a boy trying to learn to navigate life with psychic powers that are affected by his emotions, under the wing of a young conman who pretends to be a psychic. It's not as popular as the mangaka's other work (One Punch Man), but the animation for the action is on par with OPM's and overall Mob's story is a unique blend of the coming-of-age and action genres, with more down-to-earth characters (which makes them unique in this genre) than OPM. OPM has some epic moments, but not with the same emotional weight as in Mob (but it's not heavy). OPM feels like a fun comedy-action he made, but I'd guess Mob is the one closer to his heart.
