That to me seems just a matter of definition, i.e., whether you define 'sound signature' as something measured or something perceived.
It can even be more subtle than that: The visual illusion picture does contain a 'signal' (pattern of colors and shapes) that
causes the illusion of movement. So, in some sense you could argue that the visual perception of movement is actually encoded in the picture.
In a similar way you can encode auditory perceptions in a DAC output signal by (dynamic) manipulations of phase, frequency response, etc.
You could even disable these signal changes if the DAC detects a measurement signal. Such implementations do exist, like in this
test of a Denon CD player on Audioscience review: