English word choice is confusing here because voice can be "the style of speech/writing".
By changing [style of] their speech I mean the frequency, timbre, intonations, accent, nasality, etc., and NOT words, or content (which obviously can also be changed).
There are often key elements that match based on morphology of the vocal apparatus, but if I speak like a toff with a cold vs. a barrel-chested Nor'n Irelander there's little ID left.
Good mimics, those who have radio/TV shows for example, can if they choose leave nothing for a normal listener to distinguish their speech from that of the subject of their mimicry. Impressionists tend to charicateur their subjects somewhat, however.
In UK it used to be relatively common to have a "telephone choice" that would sometimes confuse even close acquaintances in to thinking you were someone else.
Another common example is when taking different languages one has a different voice.
Can't it? I have no idea but if you can train to speak in a different voice, how is that not changing your voice? Maybe I won't be able to speak like someone of the opposite gender because I don't have vocal chords like that, but learning to speak like someone with a similar body should be doable, no? Or can speech recognition systems really pick this out somehow and is there really the concept of a unique voice to a person?
One can definitely train to speak better with coaching, but your voice and most specifically your voice “ID” is unique to you. That can’t be changed.