I think it mostly comes down to how they handled the 5th gen, and to a small extend the 6th. With those two they put themselves so far ahead that they would need to do something extremely stupid to lose that position in the market. Not even the fiasco of the PS3's first years could destroy that position. They will need to do something similar to the 2013 Xbox One reveal and be stupid enough to go through with those kind of ideas, to lose their position.
When it comes to the original Playstation, Sony did everything right at the time in gaming history where it mattered most:
-They saw that CD's was the future (Nintendo did not)
-Their console was easy to develop for (N64 was too, but was held back by cartridges, Saturn was harder, though that didn't really stop third parties)
-Their console was cheap
-The console came at the perfect time, where affordable hardware had just gotten good enough to generate decent 3D graphics (N64 was too late)
-Clever controller design and revisions
-Strong new 1st party IPs (Grand Turismo, Crash Bandicoot)
-Great third party deals. They must have written Namco and Square some pretty decent checks for keeping their killer apps off the Saturn. Those two are probably as responsible for the PS success as Sony was.
-Sony had the money to do the marketing and sell the system at a loss.
When the PS2 came along, the race was pretty much over. Sure, the inclusion of the DVD drive helped Sony to snatch an even bigger market share. But I would bet that even if the Dreamcast and Gamecube had that and the PS2 didn't, the PS2 would still be wiping the floor with the competion.
I mean, from a hardware perspective, the PS2 is easily the worst designed of the generation and the DVD player was pretty bad too. None of that held Sony back. They already had the third parties in the palm of their hands and could treat them as they wanted to.
Thankfully the strong grip on the industry isn't as strong as 15 years ago. Microsoft has had periods of really good competetiveness and the Nintendo brand is doing better than it has in decades, thanks to some really smart business decisions. But none of them are really challenging Sony's position.