I didn’t have my own console in the late 90s. My brother’s friend would come round often with various iterations of Nintendo consoles and a Playstation. We would have long gaming sessions (he was unbeatable though) and I remember Goldeneye and Mario Kart on the N64 and Wipeout on the Playstation.
I can’t remember many other students having PCs when I was at Uni in the early 2000s. I don’t think this was so much affordability as a lack of interest. For me the computer was a gateway to the Web and the wider world. Perhaps it was more to do with form factor as it was a bit geeky to have a PC on the Web in the 90s but nowadays everybody seem constantly connected through mobile phones and tablets. An Internet-connected PC suddenly opened up multiplayer online gaming and no longer was I bound by the availability or interests of friends, I could play anybody around the world. I played some 90s games in the 2000s and more modern games like Unreal Tournament 2003, Counter Strike, Grand Theft Auto 3 and Half-Life 2.
At Uni, and shortly after, most of my gaming was on friends’ consoles so we’d typically play multiplayer games. I never got into console single-player campaign mode games in the 2000s. My friend was the only guy in halls who had an Xbox and we’d spend many hours almost exclusively on Halo. Later, I lived in large shared houses, and again, the choices would often be Halo 2 or Project Gotham Racing 2.