Eh I think One Piece due to its adventurous vibe. Granted I was a 15 years old teen when I first watched it, but the adventure tune like the Skypiea arc with their ship traveling to the sky seemed pretty immersive.
As for something psychological, I think it'd be manga as I haven't seen that many psychological animes. Oh actually there is one, perfect blue... That one seemed great and pretty psychologically immersive. I knew the Black Swan movie was inspired in the anime and watched it, and found the anime even better. I think DB would definitely enter in immersive stuff as well but more because I was young. HxH too, but being young played more of a factor with those, Perfect Blue was something I've watched past year. Though HxH, the Chimera Ants arc for instance, would be highly psychological immersive even as of now.
Manga tho, I think Chi no Wadachi... Not that I've lived anything similar, but I could feel the scene by scene tension, grief and such... Aku no Hana as well, a lot actually, I think even more so than Chi no Wadachi. Vagabond was pretty immersive as well, even more so towards the farming arc... The development the protagonist went through and all the emotional impact made that pretty realistic. Berserk, as well, but I think being young played a factor. Not that I imagined myself in those situations, but things like feeling the weight of each scene. Lone Wolf as well, mainly towards the last arc. I've liked most arcs, but once the story connected as a whole it was very psychologically immersive, so much so I've read the ending many times, since the protagonist's speech hit really hard. It was epic, sad and very well built. Another scene as well, though I won't mention due to spoiler. Holyland as well, in a way. The protagonist dilemma felt very well built up and relatable. Kingdom too... Not from a relatable perspective or something that touches in the sense of a problem you can feel with yourself, but I think the epicness of some battles, some grievances of the old era of powerful clashing with the new era and the struggle of acceptance of a new generation taking over being hard to grasp... And now interesting some characters can be that draw you to see the clash between each line of thinking, strategies and attitudes which all makes it very immersive... Though yeah, not so much from a psychological standpoint.
Oh, and Solanin from a physiological perspective as well, ofc. The struggle of early adulthood life and the struggle between dreams and societal expectations, the expectations you have from others... It's very real and can hit hard.