Extremely common misconception (even among developers on those platforms)

In reality, the OG Xbox and GameCube GPUs are almost identical in pixel shading capabilities (Though the gamecube's vertex shading pipeline is legitimately fixed function, but very flexible).

Despite their roughly equal capabilities, they were exposed with very different APIs. The xbox used the new-fangled "Shader" style API that Microsoft was introducing to the industry at the time, while TEV used a very extended version of the older "Texture Environment" style API that was introduced with DirectX 7 and OpenGL 1.3.

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Edit: Actually, it might be better to explain from the other end:

In a true fixed function GPU like the Playstation 2 and Dreamcast (or OG Playstation... but not the N64, which is a two stage register combiner) the pixel pipeline is limited to just one basic equation. A single texture is sampled, and that sample is multiplied with a single color interpolated from the vertex colors (which were usually derived from lights). The flexibility is of the equation was limited to replacing each input with a fixed value, and then enabling a few optional post-processing stages like depth based fog, alpha cutout and/or blending with a few fixed blend equations.

But the results from that single texel * vertex_color equation are limiting. A common technique to produce better results on such GPUs was "multi-texturing". Graphics developers of the era would render the same triangles two or more times, but with different textures and vertex colours, blending the result into the frame buffer. This was commonly used to achieve the illusion of more detailed textures, or texture based light-maps. Or the reflections on cars in racing games.

But blending in the frame buffer is expensive as it wastes a lot of memory bandwidth. The PS2 is hyper-optimised for this approach, it has the VUs which can quickly generate multiple draws of the same geometry, and a fast, embedded dram with enough read/write ports that it can do blending "for free". But in the PC world, GPUs started adding features to combine these multiple draw call together and blend the result before writing to the frame buffer. The Voodoo 2 and Nvidia TnT (Twin Texel) from 1998 are examples of GPUs that supported this multi-pass texturing.

DirectX and OpenGL provided the "texture environment" APIs that automatically used these new single-pass multi-texturing features when available, or would fall back to multi-pass rendering on older GPUs.

But the actual hardware was often more flexible than what DirectX/OpenGL exposed, though vendors supplied "Register Combiner" OpenGL extensions that exposed the full functionality (This is why John Carmack used OpenGL, so he could create optimised per-gpu render paths for each GPU). And these Register Combiners could be "programmed" to produce pixel equations that were way more advanced that what could be achieved with multi-pass rendering, as they could pass more than one value between stages. And they started supported 4 or 8 textures plus enough math stages to combine the textures.

Microsoft gave up trying to expose the full capabilities of these register combiners though the older Texture Environment APIs and introduced Pixel Shaders with DirectX 8, but they were just providing a new API for the features GPUs already had. The register combine stages were simply renamed to "instructions".

The Xbox is a register combiner with 4 texture fetch stages and 8 combiner stages and Gamecube has 16 combiner stages and 8 texture fetches (well, it technically supports 16 texture fetches, but there are only 8 sets of UV coords)