I want to say that Playstation didn't attempt to approach this angle with the PS1 or PS2. I'm not sure about that. Sega was trying to be ahead of their time and they improved on this with the Dreamcast, which nobody actually purchased. It worked a lot better on the DC and it was stock standard included with the console. I played a few racing games and Phatasy Star Online with the Dreamcast but of course it was plagued with disconnects as well.
I'm not sure about the PS1 but the PS2 at least had a network adapter. It plugged into an expansion port on the back and provided an ethernet port. I'm not sure what games supported it. I got the PlayStation over the Saturn because at the time there were tons more games for it and it was less expansive. Later, i had both the Dreamcast and the PS2 but I liked the Dreamcast more.