It can be about either the sensor's sensitivity to specific portions of the visible spectrum or about the algorithms used to create color from the monochromatic luminance information collected by the sensor. But it is almost always about how both are combined to produce a viewable image.

You can take the same raw image data from the same camera and run it through two different processing pipelines and get two different results.

Similarly, you can take the raw data from two different sensors imaging the same scene with the same lens and run each through the same processing pipeline and get two different results.

Images captured using Bayer masked sensors must be extensively processed before we have anything remotely resembling the way we perceive a scene with our eye/brain system. When you open a "raw" image you're looking at an extensively processed interpretation of the information collected by the sensor. Differences in both the camera's sensitivity to various wavelengths of light and differences in how that information is processed both have an impact on the image we see.

I've been curious about the differences in Bayer color filters between different manufacturers - do any of them try to optimize the light transmissivity for low noise at the expense of color accuracy?

They pretty much all do that to one degree or another because they all mimic the human vision system that uses multiple types of cones in the retina. Not all of the cones in our retinas respond equally to all wavelengths of light. Each type of cone responds to various wavelengths of light differently than the other types. Our brain then compares the difference in response of the different types of cones to a specific light source to create what we perceive as color.

Cameras that use Bayer masks (or the external applications that interpret the raw data from a camera's sensor) do the same thing: They compare the differences between the signal obtained from pixels masked with different colored filters to the same light source and attempt to create color that mimics what the human eye/brain system creates.

Comparing the spectral response curves of different sensors, such as those included in some of the answers at some of the links above, will reveal the differences you seek. The peak response of each channel will tell where the each filter is tuned to be least restrictive (making that pixel well 'most sensitive" to that wavelength). The flatness or steepness of the slope on either side of the peak will tell how "strong" or "weak" each color filter is (assuming the scales on the axes on the charts being compared are all the same).

But it is hard to say if one profile is more "accurate" than the other, because different humans can also have slightly different color responses to the same stimuli. One camera's profile may, in fact, more closely resemble how one person perceives a particular scene and another camera's ever so slightly different profile may more closely resemble how another person perceives the exact same scene.

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There's no real thing that equates to what we call "color" until a mind creates color based on stimuli. Light has no intrinsic color. What we call 'visible light' is not intrinsically different from non-visible wavelengths of the electromagnetic spectrum. From a pure physics standpoint the only difference between microwaves, radio waves, visible light, X-rays, etc. are the wavelengths and frequencies at which they vibrate. Those different wavelengths and frequencies do affect how they interact with various materials and energy fields, but the fundamental principles of how they work are the same.

The only thing that makes "visible light" visible is that our retinas have chemical responses to certain wavelengths of EMR and our brains interpret those chemical responses to create colors.

The colors our brain creates are based on the differences in the response that the different types of cones in our retinas have to the same light. There are colors that we perceive that do not equate to a single wavelength of light. Those colors are our brain's perception of certain combinations of multiple wavelengths. Other species can perceive the exact same wavelengths or combinations of wavelengths of visible light differently from humans. Often the range of wavelengths they perceive as visible is different. Many animals can see wavelengths humans can't and vice versa.

If, for example, we wanted to create a camera system that would provide "color accurate' images for dogs we would need to create a sensor that is masked to match the response of the cones in dogs' retinas, rather than one that matches the cones in human retinas. Due to only two types of cones in dog retinas, they see the "visible spectrum" differently than we do and can differentiate much less between similar wavelengths of light.

The chart above explains why we think our dog is dumb for running right past that brand new shiny bright red toy we just threw out in the yard: he can barely see the wavelengths of light that we call "red." It looks to a dog like a very dim brown looks to humans. That, combined with the fact dogs don't have the ability to focus at close distances the way humans do - they use their powerful sense of smell for that - leaves him at a distinct disadvantage since he's never smelled the new toy you just pulled out of the packaging it came in.

Back to humans.

It turns out that not all humans have the same number of cone types in their retinas. Some (most) of us have three. A few of us, almost exclusively female, have four. Those whose brains actually use the extra cones to perceive color are known as tetrachromats. The extra sized cone is a slightly different length most sensitive to yellow that lies between the "green" and "red" cones, which are already centered on wavelengths a lot closer to each other than to the wavelengths on which our "blue" cones are centered. This increases the amount of "overlap" between the "green" and "red" cones and allows tetrachromats to perceive smaller differences in slightly different shades of color.

In case you are wondering, the 8-bit sRGB color space is too limited to allow tetrachromats to differentiate more shades of colors on sRGB devices than the rest of us mere trichromat mortals. Human tetrachromats don't have a wider color gamut than the rest of us, either. They just can perceive more shades of color within the same total range of colors. There is evidence they can perceive variations on color in dimmer light than the rest of us, though. Other animals that are tetrachromatic can have extended ranges of sensitivity on both ends of the "visible" spectrum. It all depends upon the physiology of the cones in the retina.