Just one correction I would make is that Playstation was not the standard console for last generation. I believe that would be actually a tie between Xbox360 and PS3, or maybe a small advantage for PS3 because it ended the generation much better than XBox. I also wouldnt count the Wii here because I believe real gamers never had just a Wii. I believe most had a Wii AND a 360 or PS3.
But answering the question, I believe it is a mix of 1) consistency, 2) best tech (media being a big part of it) and 3) games. So 3 factors.
1) Consistency: PS1 was the first videogame where we had the impression was targeted for late teens and adults. It wasnt only for kids anymore. That was something that you would see mostly on pc games, not on console videogames. Although you had experiences that were mature on genesis and snes, when PS1 arrived it was just another level specially after 1996. You had resident evil, ww2 first person shooters, syphon filter, gran turismo (which was an "adult" racing game) and even the RPG's were more mature. Final Fantasy VII, Xenogears, Tactics, they were all in this vibe of something that was crafted not thinking about 12 year old kids but something that would appeal for older kids. It resonated a lot with people like me that were 16 in 1996. Games for the first time were not only for kids. On PS2 it was the same, a lot of focus on mature games, kinda of knowing that the kids that bought PS1 5 years ago were older and we still needed to keep building games for them. This kept happening on PS3 and PS4. Playstation is not a brand with games for kids, party games etc. It is for a more mature audience and if you want something colorful and full of charm for you kid you should go to nintendo. While Nintendo didnt know exactly where they were during the N64 and Game Cube eras, Sony was full on with it's message and who they wanted to achieve with their games. Even the look of the videogames showed that, it was not a just a toy for kids.
2) The best tech (at least from the average consumer point of view): Playstation always had the best tech from the point of view of the average consumer. When the PS1 was released, looking at cartridges was almost a joke. The loading times were bigger, but most people didnt care. The level of detail that could be achieved with the CD was so much better than using cartridges, full motion videos on games, better sound and textures. We knew that was the future and nintendo was stuck in time and there was no turning back, and it helped give the PS1 a more mature feeling. During the PS2 era well what there is to be said? A lot of people bought it only to have a DVD player, you could watch movies on your videogame. GameCube could have better stats but nintendo decision to have it's own media was just a bad move. The controller was such a step forward, while nintendo was using that weird format for the game cube. During the PS3 era, again a great move of going for Blu Ray something that the others didnt have, although the price was higher tha the Xbox360. And despite Xbox run multiplats with better performance, the PS3 games were not that behind, it was pretty close. But when you looked into the first party games then you would see the full power of the machine and it could not be matched by Xbox. What it lagged behind in this era was on connectivity and online services but they would mostly catch up by the end of the generation. On Nintendo side it was just incomparable, terrible online service and the games were not even HD. On PS4 era, again, better tech while Xbox became known as the least powerful machine.
3) The games. Sony made TWO very very important decisions here: first embrace the third party developers that were let down by Nintendo. This was very very important during the PS1 era because at that time there were no big first party games, maybe gran turismo. But Capcon, Konami, Square, they all embraced sony and it's media and Sony embraced them. Although not first party the fact is that you could not play those games on Saturn (lack of power, difficult to develop and low capillarity) and N64 (bad tech). There was only sony and they were smart to make great deals with those companies and those third party games were the biggest reason why sony won that generation. Gran turismo would be the big exception here cause it was first party and absolutely huge. During the PS2 era sony realised that if they wanted to keep beating nintendo they would need games so they increased the number of great first party games and the biggest addition here would be god of war. The same for PS3 with the addition of naughty dog games. PS4 with Spider Man, HZD and others. Microsoft had the chance of doing the same but they just didnt during the 360 era as they ended the generation lagging a lot behind sony in terms of first party content. There is Halo, Gear and Forza, but Halo and Gears do not even use the same studio anymore, so their list of killers first party games got small and lost relevancy during time.
Overall, this generation most likely will be won by Sony again. It will keep being consistent with its message, it will have the best tech and if not, will be very close to competition and will have the best library of first party games which have been built since PS1 and most likely will be increased during the generation. Unless they do something very stupid.