Sony definitely deserves partial credit for that but I think gaming was trending that way regardless of them entering.  That’s just a natural side effect of 80s kids who grew up on NES/GEN/SNES getting older and graphics getting more advanced thus being able to display more realistic violence. Games like Mortal Kombat, Doom and the creation of the ESRB showed that there was a growing hunger for games aimed at older audiences and that trend continued.

PS1 is the reason why console gaming is so big nowadays. It had an affordable offering and cracked markets when gaming was heavily centered around NA and Japan

Without PS1, Dreamcast and Original Xbox would have sold a bit more. Xbox could have reached some of PS1 popularity in Europe and ROW region during 360 days, but without the heavy support of Japanese developers I believe it would never get as big as Playstation is

Western development would have been pushed towards PC much earlier, with consoles being mostly restricted to exclusives from the console makers (Sega, MS and Nintendo)

I think eventually consoles would be consumed by people who want exclusives and japonese games (Sega and Nintendo), portables (Nintendo) or simply could not afford a PC whatsoever (Xbox)

Adoption of PCs would be more widespread and with the lack of standardized controllers and engines I believe PC and console gaming would move further and further away and turn into completely different markets

In the end it's about the games. If Sony wasn't to popularize the disc format with their own system, another company would.
Nintendo abandoned the idea with the N64, but perhaps a different suitor would have picked it up (SEGA?)

My favorite Japan-only game, SEGAGAGA, wouldn't have been made, as the reason behind it's development (and it's villain: Sigma is very clearly designed around Sony, not Nintendo) would be gone.

Final Fantasy VII is such an interesting part of this scenario. If it is an N64 exclusive, or at least timed exclusive, it would have to be scaled back massively. Square started the game on SNES, but an N64 version would likely lack virtually all the cutscenes, have worse sounding music, and worse details in the world map. That or the game just goes to Saturn or PC and stays as ambitious as its PS1 real-world release.
I suppose it depends how Nintendo treats Square.

The entire game uses pre-rendered backdrops which would need to be compressed by an insane amount to fit into a cartridge(and then look worse), and they'd still need to make the game smaller with less areas. Alternatively they might have rendered the areas in real time but they'd look much worse than the pre-rendered ones. 

Tempted to say Sega would still be around, since PS2 was pretty much the meteor to their dinosaur extinction event... but they were just so horribly mismanaged out of Japan. Truly, it was a cluster****. So who knows. Perhaps it would be Nintendo and Microsoft, but Microsoft really only entered because of what Sony had done. The only thing that is a certainty is Nintendo. I think they'd still be here, and still be in a favorable spot.

Without Playstation I would have stuck around with PC as primary gaming machine.

PS1 marketed consoles to 'older' gamers like me at the time and removed a lot of the stigma that video games consoles were just for kids. The lack of PS2 would also have slowed down DVD adoption.
Nintendo would likely have been less receptive to 'mature' games, GoldenEye 007 might not have been greenlit.

PC gaming would have been much bigger in Europe without Sony moving 'older' gamers into the living room. MS would have far less reason to launch the XBox as XBox was launched mostly for the battle of the living room where Sony was dominating. MS earlier attempts at multimedia PCs for the living room failed as PS1 / PS2 did it all much easier and cheaper. Instead MS could have continued partnership with Sega as the Dreamcast wouldn't have faced competition from PS2. Online gaming could have caught on sooner as that was already build in to the Dreamcast.

Very possible mature games would have stuck to PC much longer while consoles would have stayed targeted at younger crowds.

I really think the whole “PlayStation made gaming cool for older gamers” is overplayed. That’s just a side effect of 80s kids becoming teens/young adults & graphics becoming more realistic in the mid-late 90s.

For example, let’s say you were born in 1980 and got an NES with Super Mario Bros in 1986, then got a Genesis with Sonic the Hedgehog in 1991. That kid is now 16-18 when games like Tomb Raider, Resident Evil, Final Fantasy & Metal Gear Solid released on PS1. Kids who play games become teens & adults who play games.

As for graphics playing a part, a bunch of games in the late 80s wouldn’t have been viewed as kids games if graphics weren’t so primitive, some examples are Ninja Gaiden, Contra, Castlevania, Splatterhouse, Metal Gear, Altered Beast, etc.

In 92/93 we started to see games like Mortal Kombat, Night Trap, Wolfenstein & Doom make waves for their depictions of violence, not because games all of a sudden became violent but because advancements in technology made the violence more realistic.

PlayStation became the “cool” console for teens & adults but they didn’t create that market, the industry was already trending in that direction and would have continued to with or without PS, what Sony did do was capitalize on the mistakes that Sega & Nintendo were making with their hardware in that time period and became the go-to console for those games.

I don’t think this is correct at all, the PlayStation represented a complete revolution in the perception of video games. Millions of people were buying it, teenagers and older people, who had never played video games in their life. 

The NES —> SNES generation was basically flat with the leading platform actually selling less overall (62m->49m). The PlayStation and PlayStation 2 generations were the biggest growth periods ever experienced. People didn’t stop getting older, there were always new generations of kids growing old and becoming teens who played games, and being replaced with new kids, we’re now at a point where huge amount of the game playing demographs are like 30+ and 40+ people. However the overall market never significantly increased again.