Personally I modify my speech by context, just as I modify my writing. When I'm "home" (where I grew up, my parent's house) I emphasise my accent a little more, just as I use local slang.
Some people, subconsciously echo a speaking-partners accent - changing the tone, timbre, inflections, and such between conversations.
But yes, you're right that voice was strongly linked to his identity as is often the case for prolific public speakers.
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.
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.
This is known as code-switching. It's different from modifying your "voice" (although I suspect code-switching to a different pitch may kinda count).
If using a different accent (different range of frequency, different speed, different tone) isn't changing one's voice then I'm not sure what really counts? At this point it sounds like you're trying to use a non-physics based concept for voice [the parameters of sound produced by vocalisation].
Speech is so nuanced; it really seems true that 80% of what you say has little to do with the words themselves.