Exactly, which was through Square Enix. Sony created the addon hard drive specifically for that game. The online network adapter on PS2 was primarily used for that game and SoCom. There was no monthly fee's for online play through Sony at that point as PSN didn't even exist then.
There were TONS of online games for the PS2. From WWE to Tony Hawk even though it wasn't the PSN, yet. But even though Xbox charged and PS2's online was free, on paper it didn't hold a candle to the Xbox--and I didn't even have an Xbox!
And I did get an Xbox 360 in 2007 and a PS3 in 2008. The Xbox was so far ahead, it was crazy. Little by little, PlayStation added features that narrowed the gap considerably by the time the gen ended. But, that generation also saw the PS brand lose a ton of market share and Sony was hemorrhaging money to the point where we weren't even sure they'd be around.
Sony kinda had to have the bullet point of "free online" to get good pr and it cost them a ton of money they really needed at the time.