Most influential system (and why)

[BraLoD]

PS1, the NES was important but gaming took off with 3D, 2D became just a side part of it, both are important and the PS1 stablished both things (3D > 2D, but should have both).

Also, the Dualshock is still the configuration of every controller out there, every single controller is a Dualshock configured by a different company.
Grips/handles on both sides, analog sticks on both sides, 4 shoulder buttons, 4 face buttons (SNES), dpad (NES), built-in rumble, pretty much every controller out there since it is built around it.

The PS1 defined the way people on consoles play up until this day.

Also it was responsible for creating a 3-axis power were America, Europe and Asia were all very important at the same time, making console sales pop the last bubble and become mainstream worldwide, for all ages.

There is nothing remotely comparable to the PS1 out there, the NES helped gaming get back on its feet in the 80s, the PS1 made it be what it is up until this day, for over 30 years now.