NES, most. But second place, can't believe I am saying this, is the Switch. I mean the sales are crazy. It clearly demonstrates convenience over high-end graphics. The S2 is off to a great start. MS is working with Rog for a xbox handheld. Rumor has it Sony is going hybrid at some point as well. Feels like the Switch has forever changed the market. I still think NES is first, but the Switch would be second on my list.

Virtual Boy.

NES and it's not even close really. The entire modern game market basically traces back to there.

PS1 is overrated, it just took the 3rd party support that Nintendo already had for 10 years ... where do you think Final Fantasy and Metal Gear and Dragon Quest and Castlevania got popular? Capcom was already one of the most famous devs because of the NES and SNES, it's not like they just popped out of the ground with Resident Evil. Sega was already marketing towards older audiences/teenagers too, so the whole "Sony started marketing at non-kids!" thing is bunk too, Sega was doing that a generation prior already. 

Sony just got fortunate that Nintendo made an extremely stupid decision to use cartridges and the double whammy of Sega basically killing itself, leaving the PS1 with a wide open path. If the N64 used CD-ROM, I doubt the Playstation wins that generation at all, they'd have been no.2 and a distant no.2 at that. Even the PS1 OG controller design is basically a rip off of the Super NES controller and then they had to see Nintendo use analog + rumble on the N64 to copy that as well. They just copied a lot of the Super NES and added in more Sega style marketing (right down to their Sonic knock off of Crash Bandicoot). 

PS1

Me thinks NES era gaming was still seen as mostly toys for kids spend time. If things were kept as they were during NES era I don't think gaming would be as big as it is now

Simply put, newly games based on NES design are "retro" i.e. trying to capture older audiences with nostalgia

Games based on PS1 philosophy in other hand still push gaming forward, trying to capture new audiences

Conversely, 2D games trying to capture new audiences/pushing gaming forwards seems to be mostly based on old PC games, not NES games. They are generally simulators, CRPGs, 4x Strategy, RTS, Immersion Sims, and even Roguelikes

I honestly think NES is an object from the past, but this is me lol

I'm gonna throw in the OG Xbox as far as setting the template for basically every modern console now; standardized parts with PC-like architecture, (as opposed to the more bespoke custom stuff in say PS2) with a built-in storage drive.

In my opinion, the Atari 2600.

Even if there was the first gen before it, to me the 2600 was the true start of gaming.

- Atari 2600 - for making interchangeable cartridge format popular (Fairchild VES/Channel F being the first), thus making software business on open (non-willingly, since they lost in court to Activision) hardware platform (for all the good and bad) a viable option, effectively kickstarting 3rd party developer console industry.

- NES - for popularizing D-Pad (no, Nintendo didn't invent it, it was Tiger Electronics) and making it standard for consoles, as well as for introducing "Seal of Quality" and all the related restrictions, as different type of approach to 3rd party developer business model that managed to return fate in home video games to NA market after the crash of '83

- OG XBOX (AKA Dreamcast 2)- for being first console to have architecture that will set the blueprint for how consoles are built in later generations, introduced HDD as standard, thus opening console market to more demanding games, as well as for making first unified online gaming experience via XBOX Live in console market.

I feel it's a bit too soon to tell for any other console after that.

The answer is NES, but SNES had the most influential game controller of all time.

Excluding Gen. 1...

I understand "influencing" to mean the act of establishing "industry standards"... Abandoned standards are of lesser relevance; those that persist are of greater influence.
Two systems seem like obvious choices to me.

PS1, Why:

Controller:
It evolves the basic design established by NES/SNES, being the largest individual contributor to modern controller design, with contributions such as:

1 - Responsible for popularizing the use of analog thumbsticks with potentiometer technology, a standard used throughout the entire industry even today.
2 - First to use pair of analog thumbsticks, the standard of all modern controllers
3 - First to use L3 and R3 buttons on the analog thumbsticks.
4 - first to Add of 2 extra buttons on the shoulders. (L2-R2, ZL-ZR, etc.)
5 - First to add internal motors in the controller grips for vibration.
6 - Ripping off Evolving the design established by the SNES. It set the standard for modern controller ergonomics, allowing for agile transitions between Analog Thumbsticks/D-pad/Face Buttons.

The controller is so absolutely influential that its configuration (established in 1997) would be able to play most games on the market in 2026. The "standards" it established continue to be universally followed throughout the industry.

It popularized media with the exclusive function of storage:
Many tried to use CDs, all failed. Playstation 1 sold over 100 million units and established digital media as the industry standard. Digital media has become so utterly dominant that many cite it as a key factor in the original Playstation crushing the N64.

Media and entertainment functions:
A modest beginning, the Playstation served only to listen to music. But this is where the idea that consoles can serve other purposes besides gaming began.


Xbox 360, Why:

1- XboxLive:
For years, Xbox Live set the "standard to be followed among consoles." The entire industry followed their example, undeniably. DLCs, digital game purchases, subscription services... it all started here. 

2- The first modern console:
User interface gained a central position on consoles, apps, game lists, etc. all of this began with the 360, standards maintained to this day.
Playstation OG was the "Cd music boy" - Xbox was the "Netflix and other apps" Man. 

3- The controller:
1 - Triggers, shoulder buttons were a thing of the past! The trigger was indispensable for playing Halo, Battlefield, Call of Duty, etc.
2 - "Xbox" button - The button that took us to the UI, later copied by the entire industry.
3 - Wire is a thing of the past. It was the first to bring wireless control as a standard.
4 - Made for adult hands. (This is a personal opinion, fell free to ignore it, but... I feel that before the Xbox 360 controllers were designed for small hands.)

Now here is my take... 

If your main focus is OS, Network and Services as part of the console > Xbox 360 is the most influential, it shaped modern consoles.

If your main focus is hardware  > The original PlayStation is the console that set the most standards in terms of hardware, in especial, the controller and the disc drive that popularized the use of Digital Media.

The most influential is usually the oldest, because everything after is built on top of it.