Does the "color" come mainly from the preamp, or does the power amp also produce "color"? Or is it pretty much true that a power amp is a power amp is a power amp?
Yes, power amplifiers can also produce color. Tube amps are famous for that, many solid-state amps designed to mimick them.
Basically there are two schools of thought for designing power amplifiers for musical instruments, neither any better than the other. Just different.
First aims into non-coloured re-production of the input signal. You could basically refer to such design as HiFi amplifier, since its goal is producing the sound accurately, without distortion, excessive compression, or equalisation of the signal. What goes in, should come out. Such amplifier typically features moderately high power and headroom, but if its is ovedriven distortion characteristics are typically abrupt and harsh. The typical application is working as a transparent amplifier for external signal sources. If signal colouration is preferred it is introduced externally (e.g. in the preamp).
The other school of design wants to integrate a signal processor and a power amplifying unit. Basically it aims into creating a power amplifier, which deliberately colours the signal in certain way (e.g. sounds like a tube power amp). This colouring can be equalisation effect of running a high output Z amplifier to a reactive speaker load (high and low frequency boost), and/or "designed-in" distortion introducing additional harmonics to the signal, usually due to signal clipping in ovedrive conditions. Headroom may have been deliberately lowered to generate "softer" and more gradual transition to full clipping ovedrive.
Since its a power amplifier unit, it may be beneficial that such effects are moderate, though. For example, a power amp usually is expected to perform somewhat more cleanly than a plain "fuzz" effect. Equalisation and clipping limiting of the signal (with nice timbre of distortion) can be nice musical effects, but not often when dosed too excessively. In addition we may need to ponder the fact that not all these effects fit bass amplification as well as guitar amplification. So even smaller doses may be preferable.
But yes, we do have power amps that are deliberately designed to colour tone. Many solid-state musical instrument amplifiers are designed to feature high output impedance and we see a growing trend of soft clipping or tube emulation architectures being added to musical instrument power amps. Peavey, Hughes&Kettner, TubeWorks, Ampeg/Crate, Quilter, Line 6, etc. List is long if you count solid-state amps of all type.
I don't really know about class-D -based units though, but at least Quilter amps feature deliberate tube amp emulation integrated to the power amp section. There's an important catch: "Tube emulation" of Quilter's is independent of what class the amplifier works in. It takes any transparent power amplifier (a class-D module fits the job perfectly) and then adds external circuitry around that building block, which in turn colours the signal. Any amp design can do this, so its kinda pointless to try to evaluate performance of an unknown amplifier based only in knowing in what class the amp operates in. But I guess the point is to say that designing a class-D amp that colours the signal is just as trivial as designing any other amplifier to colour it. Class is not a limitation.
This not even dwelling into specific architectures of different class-D circuits or predesigned gate driver chips in general. For example, how will the circuitry handle over modulation? There are designs that hard clip nastily, and then there are designs, which go to nice soft clipping mode. All this covered by class-D category, as it is so broad. If we start to evaluate "colour" of an amplifier we quickly find out that not all class-D amps and modules sound alike.
