I realize I'm biased having experienced this era at my most influential (as another user easily defined it as ages 12 - 22), but this was definitely it for me. I only had a Gameboy before I finally had a PS2. The big mascot character games of this console were formative for me. Jak and Daxter, Ratchet and Clank, Sly Cooper. Kingdom Hearts and Shadow of the Colossus were everything to me. Tons of other huge titles made this generation.

But it's the weird little games that I think about fondly. Katamari became a franchise, but it was just a funny novel idea when it dropped on the PS2. Kya: Dark Lineage, an adventure/fighting game absolutely packed with fun ideas from a studio that just made racing games prior. Magic Pengel - basically DIY Pokemon - was pretty much everything I wanted in a game. Even Eye Toy, which completely sucked and barely worked, offered a new way to play games.

Things were just different then. I think it was maybe the last time we thought of games by their budgets. Most titles were what we would maybe call AA these days, something that almost doesn't exist anymore. Where indie games didn't exist yet, but small studios were prolific. For me, any game that let you run around as a fairly detailed 3D character in a cool setting was magic to me in a way the flat, pixelated worlds on my GBC never were. The worlds in my PS2 were believable.