Playstation established domination with third party superiority.
Only Gran Turismo has been a long time franchise. Infamous & Uncharted series are decent thought.
I didn't like The Last of US & Horizon , even thought i gonna try them again in the future.
I don't like God of War and I haven't tried Spider-man, maby in the future.

Third parties played a key role in PlayStation's longevity and success, but first party was also just as important in building PlayStation as a brand as well. IPs like Parappa the Rapper, Crash, Spyro, Jak, Ratchet & Clank, Sly Cooper, Ico/SotC, Ape Escape, MediEvil, and others are still beloved, whether they had a new game in recent years or not. All these games still helped shape the PlayStation feel and aesthetic just as much as third party games like Metal Gear Solid, Resident Evil, Final Fantasy, and Gran Theft Auto.

I would question whether Spyro and Crash were 1st party? 

If you remove those games, then definitely those other IPs are nowhere near the importance as a Final Fantasy, Tekken or MGS in shaping the brand. Jak and Ratchet deserve a shout out but Ape Scape, Parappa and Sly are just very niche by comparison. Parappa/Ape maybe reflect some of the quirkiness of the japanese arm and of the specific time (90s) Playstation was born from, but we shouldn't give them equal weighting in brand identity as GTA/Final Fantasy.

Edit: I'm speakin more broadly at the notion these brands did jus as much heavy lifting. Subjectively they could of course be someone personal association.

The already mentioned God of War series was what came to mind directly, even If have not played any of the games in the series shame on me.

Games from team ico is also something I associate with PlayStation as well. Not because of sales numbers but because that type of experience could not be had on other game systems.

Sony published the original games on PS1, and their respective developers (Naughty Dog and Insomniac) are now owned by them. So I say they count.

Nintendo has a pretty clear identity. Mario is their mascot and many of their IPs are cartoony. Shoot, even Xbox has somewhat of a clear identity. Lots of multiplayer non-cartoony games like Halo, Gears of War, Forza, etc. PlayStation is harder to peg down over nearly 31 years.
Astro Bot (formerly Sackboy formerly Crash or Spyro) represent the more cartoony side. IPs like The Last of Us, Uncharted, God of War, and so on represent the M-rated cinematic games. And since those games have been Sony's bread and butter since around 2010, I suppose they are the most PlayStation like games.

PlayStation has always had a cinematic feel as a brand. I thought that was pretty well known even back in the PS1 days. Games like Metal Gear Solid, FFVII, Parappa, Gran Turismo, even Crash and Spyro felt more cinematic than the arcade style games on Nintendo and Sega consoles at the time. 

I think the difference now is that that cinematic style stated to become more and more homogenized over the years. 

Returning to this topic, I think you can break the PS style into two eras, pre-Last of Us, and post-Last of Us. Pre-Last of Us PlayStation felt more like a music label. Cinematic, but it came in a variety of flavors. Parappa was an interactive music video, games like Ratchet and Sly were interactive DreamWorks movies, InFamous an interactive superhero comic, GoW and Uncharted interactive summer blockbusters, etc.

And then there's the PS style after The Last of Us. The Last of Us was the birth of the "modern" PlayStation formula, and most PS first party games afterwards were cut from it's mold in some way. Third person, over the shoulder, gritty, and much more somber. Horizon, Ghosts, UC4, Death Stranding, etc.