Totally feel that same way about DKC2. That tiny fragment of a screenshot you posted got to me.
This is a pretty preliminary response off the top of my head but I'd instantly think of:
Pokemon Red Version...when I sit down and seriously think about that game I get a little teary eyed. When I remember first leaving Pallet Town with that music playing. It reminds me of simpler times. It reminds me of specific friends I had when I was 5 or 6 years old who I haven't talked to in at least 15 years, yet I wonder how they're doing now.
Final Fantasy 6, 7 and 9 give me that same feeling. When springtime comes every year I throw on the FF9 soundtrack and go on long walks and it makes me happy. During the early Winter and closer to the holidays I listen to the FF7 soundtrack. I think it's because I associate those times of year with when I first heavily played those games. Any time I find to replay those games winds up being extremely rewarding and helps me learn more about myself.
Turtles in Time will always be there for me and is always a game I can pick up and play to forget life.
Super Mario World is pure happiness to me. Every sound from that game. Everything just makes me happy.
Strike Gunner on the SNES as well as Super Bomberman 2 really take me back to happy times way before a certain friend of mine got brain cancer. Playing these games two player nonstop with Doug and Tyler (the other Tyler) even well into middle school when SNES wasn't "cool" anymore. I could kind of lump Rampage 2 into that category, but that isn't a game I feel is always "there for me" so much as giving me the same type of nostalgia.
Smash 64 is 100% on this list though
On that previous note...I think I will add Streets of Rage 2. EgoOrb/Tyler will know exactly what I'm talking about. This doesn't hearken back to my time as a small child so much as it does to 6 or 7 Summers ago.
Super Mario RPG for sure too...I'll have to do a full write up of this at some point, but that game had a huge impact on my life.
In just about every single example it winds up being the music that really gets me and not just the colors of the sprits or the aesthetics of the game. There is also no coincidence that most of the ones that come to mind are SNES games. SNES was my first console. My dad trash picked an NES for me when I was young, but it stopped working soon after and my mom gave away the few NES games I had. So I had to rediscover my love for the NES a few years later. But SNES never left my life.