the ps2 was designed as something more than just a video game console. it was the start of sony's vision to dominate the living room (otherwise microsoft would have ignored the console market as a hardware maker). as such, it did one really big thing to make it more than just a games player, and that was play dvds at a time when other dvd players were really expensive. a lot of the early appeal to the ps2 was that it was a dvd player, but it was a games player too... so the value proposition was there beyond it being just a games machine.
it was a very mainstream machine from the start, and that's evidenced by the lack of crazy-major sellers for the period leading up to gta iii's release (which is really what defined that generation in terms of software, at least from my perspective).
The PS1 played music CDs, towards the end of its life the system could also play movies of some format from memory. The idea of consoles being more than just games machines and rather entertainment hubs had begun in earnest already. And the idea that the reason the PS2 managed to sell being because it was a DVD player that also played games, rather than as a game console first and foremost seems somewhat revisionist. The system was where Final Fantasy would go. And Metal Gear would go. It was where Gran Turismo, which at the time was more significant than it is now, would go. There wasn't bizarre talk of it's CPU being some sort of nuclear missile risk because the system could play DVDs. And people didn't defer from buying the Dreamcast because they wanted to watch movies. And it's
initial demographic focus, and that of the software on it, doesn't particularly strike as different to that of the current systems - young males.
The 360 and PS3 followed the same modus operandi, providing a central entertainment box, with one doing it well and one doing it at a crazystupid price. The PS4 and XBO are again doing the same thing.
Conversely, the Wii from it's outset was focused on audience expansion rather than fighting for the traditional market. It brought a new, accessible, simply and intuitive way of playing games that resonated with people beyond the young male market. It had software that wasn't specifically designed towards those consumers, and in fact was specifically aimed at other markets, again from the very outset. It sold abnormally well, abnormally fast, and the software that sold well on it tended to be titles that appealed to a different audience base. It also saw decline abnormally fast. It wasn't a "traditional console" and its rapid expansion and collapse don't lend well to interpretation of the health of the fundamental or traditional underlying market for console games.