Consider that DAC's are 'flat' within the audible range. Even for 44.1kHz. Depending on the used filter there can be a small ripple.
That ripple is much smaller than the 'adjustments' made in a studio for each individual track (instrument or voice) if they think it sounds better that way. We are not talking of adjustments of 0.1dB or so.
A DAC does not treat 'overtones' differently than fundamentals.
The purpose of a DAC is to 'translate' a digital 'value' to an analog voltage of the same (or extremely close) value as the 'described' value.
When you are a young person then yes one can hear differences (also in blind tests) between certain DACs and even between the same DAC when specific filters are used. When DAC A has a slow filter and there is a roll-off in the upper treble of a few dB then this can be quite audible.
For older folks this may well not be the case.
This is not because DACs differ but because filters differ within the audible band.
In most cases
where the filters adhere to the sampling theorem and there are no high distortions (tubes for instance) the perceived differences are generally caused by
incorrect level matching,
too much time has passed between swapping out DACs or
knowing what DAC is playing (sighted test).
Then there is the often mentioned 'well music is not the same as test tones' argument. And its not. When looking at transducers this certainly is the case. Not so with DAC's. They just put out a voltage equivalent to the value they should be producing and seamlessly 'slide' to the next sample value they must produce. There are ways to test with music called 'nulling'.
@pkane wrote lovely software for this. The method has its challenges but it can show the actual differences between 2 DACs using music. Not surprisingly the effects (when well executed using a good ADC) are not unexpectedly different than those that can be shown to exist when using test tones.