I have the same feeling about the first games I played "seriously". You know, some games you know of and have played as a kid, but some game was the first game that you consciously invested time in to finish it. For me it was some super weird Shrek-themed clone of MarioKart on the Nintendo GameBoy Advance. As a kid I really didn't see it as a MarioKart clone, just a fun game I wanted to finish entirely.

I absolutely forgot about it and only recently saw some YouTube video about what a weird game that was. Seeing that game again with its soundtracks and weird sound effects and clunky mechanics really triggered that exact feeling of a "connection" that you speak of.