Seems like a great technological opportunity for someone to figure out.
Recently I advised someone not to buy granite countertops sight unseen, since the way they look in person is so different from the way they look in a photo - a photo completely misses the reflections, iridescence and other three-dimensional aspects of the actual product.
I didn't mention them since they have nothing to do with synthetic vs natural dyes though -- they're properties of whatever materials the dyes become part of.
That's actually a significant part of why "accurately" photographing clothing for online shopping is so hard. Getting an accurate hue is usually fairly easy with good lighting and proper white balance, and gray cards let you calibrate brightness. But the reflective properties of fibers mean that e.g. a shirt can look significantly different depending on what angle the light sources are at.
I guess that is what I'm getting at, that today's conception of what a "dye" is, is that it is a substance which is added to a material not changing the reflective, translucent etc properties of that material (which is itself probably a simplification - I can't imagine dying a fabric doesn't change these at least a little bit).
What I'm suggesting as an advancement is perhaps not called a "dye", but nevertheless would be a new way to not only change the color of the target object, but also the translucence, iridescence, reflectivity, etc.
I know there is already stuff in this area - there was some laptop case I saw recently that had some kind of micro-texture on it (the opposite of a polish, I guess) in order to make the black color of it more matte (if I recall correctly). Likewise, anti-reflective coatings and privacy screens seem like a ubiquitous, primitive example of what I'm talking about. I'm sure there are others.
Long answer: there are mainly three factors that contribute to inaccurate color reproduction.
The first is limited color gamut. That's not really a factor in dying fibers though -- we have plenty of synthetic dyes for things like bright orange, fluorescent green, etc. Just visit any athletic clothing store. And while yes nature has all sorts of shiny bright iridescence caused by diffraction etc., those properties are destroyed as soon as the material is smushed. You can't turn gorgeous insect wings into dye.
The second is that while our eyes perceive only RGB in response to a wide spectrum, there are an infinite number of spectral patterns that we ultimately perceive as e.g. a certain orange. An orange dye from one source (natural or synthetic) will have a different source spectrum from pretty much any other orange dye (natural or synthetic). BUT it's pretty easy to match the color we perceive perfectly for a single lighting condition, e.g. sunlight. But this brings us to:
The third factor, which is the spectrum of lighting. The spectra of sunlight, of fluorescent light, of LED light, is all different. The color we perceive is essentially the spectrum of the light source multiplied by the spectrum of the dye. Therefore when we change the lighting spectrum, this can change the ultimate RGB color we perceive -- so that two orange dyes that appeared literally identical in sunlight now appear subtly different under a fluorescent light.
So it is true that different dyes behave slightly differently under different lighting conditions. But there's no better or worse here, and there's absolutely no "peach ripened by a tree" or "how it's supposed to be". That's complete and utter nonsense.
The solution for good color matching isn't dyes from natural sources -- it's balanced, full-spectrum white lighting.