I had both a 360 and a PS3 growing up, but really, this time period was around when I got reeeally into looking into retro games on the internet. I started trying out fan translations of old Fire Emblem games, found a treasure trove of amazing titles that came out before I was born, and I also got absurdly into Phantasy Star Online: Blue Burst. I kept up with major Nintendo releases on DS and Wii, but I can count the number of PS3/360 games I played back then on two hands.
360 highlights-- Eternal Sonata, Blue Dragon, Sonic 06 (yes, really)
Eternal Sonata is a game I really need to revisit and finish. It inspired what became a deep passion for Chopin's music and the romantic era in general. Blue Dragon I'd also like to revisit-- I got to the last disc as a kid and soft locked myself on a timed sequence that I was underleveled for. Really sad.
Sonic 06 came out during a period where I was absolutely obsessed and I overlooked a loooot of shit. I fully completed the game twice! Still think there's potential in it, but hahahahaha I was blinded by love.
PS3 highlights-- journey, Infamous 1 and Infamous 2
Journey speaks for itself... That game is magical. Infamous I'd LOVE to see a modern collection of. I found these games super fun back then (I got the first game as a freebie from Sony during the huuuge PSN outage period). This series is also a big reason I got a PS4, which I then went on to play waaay more than PS3.
One other thing I really enjoyed on PSN was the retro catalogue and PS1 backwards compatibility. I got a lot of mileage out of this-- it was my first time playing Xenogears and FF7.
Now, in the last five years or so I've spent time revisiting this era... There's a ton of great stuff I missed. I just started Metal Gear Rising a bit ago and am loving it. I view it as a cool thing that there's an amazing era of gaming that I have barely scratched the surface of, because I was so busy looking into the past during those years that I missed some of the great stuff.