Depends on the game, it's not necessarily nostalgia, you can get emotionally attached to characters and the story.

I was thinking about this recently, as well...only different.
I don't really care about movies. My comic book collection has gone to shit. I don't have many action figures. Clothes, etc. They just don't matter. Meanwhile, I can't bare to part with any gaming hardware. I haven't traded in even the worst games since like 2010. I don't know why. I'm not trying to collect them. I don't even like to delete digital games. Something about owning (hoarding?) games feels right and losing one feels like a huge loss.

Nothing weird about it, you are bound to get nostalgic towards things that brought you some kind of positive emotions

ICO has a special place in my heart as I played that with my then long distance girlfriend (now wife) when she came to visit for 2 weeks. We didn't finish the game then and I waited with the rest of the game until I moved over to live together. It was the first game I bought after moving (left the PAL version behind, bought NTSC version) to finish together. The game itself is very good as well and still brings tears to my eyes at the end credits.

Valkyria chronicles I played with our first son on my lap who had trouble sleeping. He was only quiet and able to sleep on my lap and Valkyria chronicles was perfect as a turn based game to play while holding him.

Journey is all on the game though. I played through that 20 times at least. The music, the loneliness, then finding someone to basically go through life with, all within 2 hours, amazing.

SNES era games give me that feeling, as well as OoT!

Right?! Ah the simpler times, no worries except "I hope mom let's me play long enough to finish this dungeon today"

I have the same feeling for Demon’s Souls and Dark Souls, even though I first played them in 2014. I think it’s because of how unique they felt at the time compared to everything else. These days gaming is too homogenized, so I think your attachment to these older games could be due to how they feel unique since they came from a time of heavy experimentation.

3 words: Ocarina of Time

I was going to dismiss this as nostalgia, but since you only played these games in 2015, then I think it is something more.  You are really looking for gameplay/experience that is like these games.  I can relate, since I barely had any Genesis games back in the day, but right now I am playing through a lot of different games on the Genesis that I never considered before.  What I'm really looking for are good 2D action games, and the Genesis has plenty, but modern gaming has few.

PS1 was a great console for long story based games.  Maybe you are looking for more games like this?

The Zelda series as a whole does this for me.

Also been playing the Switch port of Donkey Kong quite a lot lately. That was the first video game I ever played, and it was at a grocery store when I was 3 years old. Still just as good.