Playstation a 3D first console
I think that is a bit of an overstatement, and probably part of their marketing ploy. From a hardware perspective, the PS1 did have a coprocessor that could do matrix calculations, which were useful for the 3D projections and transformations (scaling and rotating) 3D coordinates. IE converting vertices in an object to rotate / scale it in world space, then project it to camera space, etc.
That is helpful, but really, processing the vertices is not the majority of the work required in 3D rendering. It's the rasterization of textures. That is mapping a texture to a polygon with arbitrary texture coordinates. The PS1 had no hardware for that whatsoever, and it was done by their software library.
However, because the CPU has to do it, and there wasn't even hardware-level floating point math (relegating them to fixed point math), they had to cut corners. There is no perspective correction on the texture rasterization. That is why the textures warp and do really weird things as you move, especially for polys close to the camera that you are looking down (IE the perspective changes a lot along the length of the poly as it goes further from the camera).
If you want to claim that the PS1 was "3D first" in architecture, then that would be in flat-shaded polygon rendering only, which the GPU could do, which isn't at all what games ended up doing - they're all texture mapped.
Maybe saying the PS1 was 2.5D first is more accurate. :)