With typical output transformers (common with tube amps) speakers with wild impedance swings (and phase) present a challenging load to the amp and fidelity can suffer.A major factor contributing to this, and what is probably the most major factor in many cases, is not the output transformer itself, but the interaction of the output impedance of a tube amp with the speaker impedance variations that you are referring to. In contrast to nearly all solid state amps, most tube amps have output impedances that are a significant fraction of speaker impedance, usually somewhere between a large fraction of an ohm and several ohms. That in turn causes the voltage divider effect to have significant effects on tonality, to the extent that the speaker’s impedance varies as a function of frequency.
In the case of McIntosh solid state amps which use autoformers that particular effect is essentially negligible with most speakers, because as a consequence of being solid state their output impedance is much smaller than the output impedance of most tube amps. (Although that certainly does not mean that an amplifier having low output impedance is necessarily the best match for a given speaker, in terms of tonality). For example the MC302 has a specified damping factor of "greater than 40," which for the 8 ohm tap theoretically corresponds to an output impedance of less than 8/40 = 0.2 ohms.
Can the same be said of autoformers? Or, are these electrically different enough from transformers as to not present the same issues?
While as I’ve said the effects you appear to be asking about are usually not due to output transformers themselves, autoformers do have significant advantages relative to output transformers in typical applications. One is that autoformers don’t have to be designed to handle significant amounts of DC, while the output transformers in tube amps must be able to do that. Another is that in typical applications an autoformer just has to provide a small transformation ratio between input and output voltage, current, and impedance, while the output transformer of a tube amp usually has to provide a much larger ratio. Those differences mean that a well designed autoformer will tend to have fewer sonic side-effects than a comparably well designed output transformer.
Regards,
-- Al