Is it accurate to say synths have timbre?
an identical note played on piano versus a cello can epitomize timbre whereby the material and playing method of the instruments can be heard to be distinct.
We do not hear the material or the playing method of an instrument directly. We hear vibrations in the air that are set in motion by the instrument. But those vibrations can be recorded, for example on a wax cylinder, a vinyl disk, or an electromagnetic tape, or as a stream of digital data. When we use that recording and an electromagnetic speaker to reproduce the vibrations in the air, we hear those sounds without the presence of any materials or playing methods, yet we can still identify a cello or a piano.
A synthesizer, whether digital or analogue, generates the signal needed to drive a speaker so as to create a particular sound. The very core of sound synthesis is to manipulate timbre, whether to mimic well known timbres, such as those of a cello or piano, or to create novel ones.
The basic theory underlying this is that of harmonics. A vibrating string or column of air has several modes of vibration, each an integer multiple of some fundamental frequency. The relative strength of these modes of vibration determines the timbre; for example, the clarinet lacks even-numbered harmonics, giving it its unique timbre.
While a natural tone may be analyzed into its harmonic components, that is, viewed as the result of adding a series of sine waves, so may an artificial sound be synthesized by combining a series of sine waves. This principle underlies the technology of the tonewheel organ, perhaps the best known of these being the Hammond organ.
More generally, this is called additive synthesis. There is also subtractive synthesis, which takes an acoustically rich signal and processes it, for example through various filters, to arrive at a more complex sound than would be practical with additive synthesis.
Sound synthesis, especially in its earliest incarnations, was not particularly faithful to the sounds it was seeking to recreate, typically because it employed a limited number of harmonics or a limited degree of signal processing. As the technology improved, so did the fidelity, but the distinctive sounds of the earlier "limited" efforts came to be valued in their own right, particularly as they came to be associated with various styles or periods of music. Thus, the Moog synthesizer is particularly reminiscent of the music of the late 60s and 70s, while digital frequency modulation synthesis is perhaps the most significant defining element of the 1980s sound.
my hunch is that there actually is something going on to enable a distinction in terms of timbre, maybe taking the form of: gauge of wire, heavier components, etc.
As I hope I explained above, it is the technique of the synthesis itself that defines the timbre, not the components of the synthesizer (though the characteristics of the speaker will be significant, of course, because its frequency response will dictate the vibrations it creates in the air, but also because it might be a Leslie speaker, designed to create vibrato).
Strictly speaking, for synths, and electric instruments in general, is there a timbre dimension to talk about? Why / why not?
I'll answer that with a couple of quotes from Wikipedia. First, the lead sentence from the article on subtractive synthesis:
Subtractive synthesis is a method of sound synthesis in which partials of an audio signal (often one rich in harmonics) are attenuated by a filter to alter the timbre of the sound.
Finally, the lead sentence from the article on additive synthesis:
Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.