I feel like this doesn’t address the OP— we’re talking about how one system influenced another, not how it changed the industry. I agree that PS1 is revolutionary for how mainstream it made gaming, but that’s irrelevant here IMO.

OT: My answer would be NES. It set the standard in virtually every way. It saw the creation/realization of almost every genre (platforming, action adventure, fighting, strategy, etc.), it set the standard on how to develop a gaming console controller (no more joy stick; just a couple of buttons and a D-pad), it was truly the first console to stick-the-landing in jumping between console generations (8-bit to 16-bit; Sega and Atari’s Achilles heel), it revolutionized the standards by which third-party developers could develop for a gaming console (no more third-party consoles capable of playing NES games; established exclusivity of NES games working on NES only) (no more E.T. nor Atari Pac-Man tier releases; quality control), etc.

The NES laid the foundations for console gaming. No other system even comes close to matching what the NES did for future hardware.

Ps1, took gaming to the casual masses.

No singular console tbh. Fairchild Channel F was the first to use interchangeable ROM Cartridges. Atari 2600 made it popular. NES gave us the D-Pad on consoles and many innovative series and many of the most popular series began there from Nintendo to 3rd parties. Nintendo Seal was an innovative idea. Vectrex gave gaming the first Analog stick. Genesis helped pioneer arcade quality ports to home console and from then on until they died out every console aimed for the same. Neo Geo perfected it. In Japan Mega Drive introduced the first dual analog controller and shoulder buttons. N64 popularized the analog stick and Mario 64 set the blueprint of designing 3D games around analog. PS1 took SEGA's marketing concepts with Genesis and expanded on them to help make PS1 a mass appeal product. PC Engine CD, while an add on is the first to introduce CD Rom games. DS and Wii brought in the audience that would eventually become the main demo for mobile games. SEGA Nomad well ahead of it's time influenced future systems like Switch and Switch popularized the form factor. Game Boy made portable gaming both a non luxory but mass appeal in audiences of all ages but also quality of games. Game & Watch introduced dual screens. Atari Lynx for color screens on handhelds. Dreamcast set in motion what we know as online gaming for consoles today. The first console MMO. First console with a Camera. Xbox first console with a HDD built in. Xbox live. Sony and MS would then include it in all future consoles. PS1 popularizing dual analog sticks which would become industry standard. Wii standarized motion controls in future systems from Sony and Nintendo themselves. Forgetting a few I know.

What about ps2? Xd

It wasn't innovative. Like SNES wasn't. it was just a better version of the previous system. BUT BUT DVD! Disc based systems playing other media was not new. PCE-CD played audio discs. CDi/3D0 played VCDs. Nor was it the first gaming machine that could play movies on disc. Pioneer Laser Active. Unless we count Halcyon but those were FMV games in 1985. 

I don't see how the answer is anything but the NES. I mean it saved gaming. It set standards and expectations for home consoles for generations to come.

I think most influential were Arcade Games. A whole generation of kids learned Video Games from Arcades and it was the crowd that later bought home consoles.
If i must choose something else it must be Atari 2600 i guess (the first console with significant sales 30 million)

It's difficult to argue against 2600, NES and Playstation 1. They were each a major stepping stones, creating industry standards. The GameBoy is of course also up there, as the most influential handheld. N64 deserves to be mentioned for the analogue stick (though the Saturn might have introduced it first technically??) and especially for some of it's games that definitely were templates for generations of games (Super Mario 64, The Legend of Zelda: Ocarina of Time, Mario Kart 64, Smash Bros., 007 GoldenEye).

I would also like to add Xbox 360:
- It set many of the standards we expect from console gaming today, be it online gaming, digital storefront, achievements, pc-integration for accessories, etc.

With how large chunk of the "gaming" space it occupies. I guess the darn smart phone with google- and app store needs a mention. It brought (some sort of) gaming to the large mass. And a lot of the modern finance methods for the industry with adds and micro transactions got (un)popular there, It made digital media not only an option but the only way to play and I think it is clear that the industry as a whole is moving away from physical media.
Influential. But not maybe in the direction core gamers want.