The use of effects and pedals is a huge deal even above and beyond the model of guitar. You mention The Edge, famous for his use of pedal effects. Tom Morello and Jack White are others with unique effect choices.
The bass at the beginning of “Seven Nation Army” isn’t a bass at all. It’s White using a pedal effect to bring his guitar down an octave.
Santana is a master of the hammer, slide and bend. Bends give him that “crying guitar” sound. A lot of his distinctive style comes from those techniques and his songs are a lot of fun to learn. A good example of all three is in the opening riffs of Black Magic Woman:
Gear is clearly a big part of it. But just like your actual voice, your guitar voice will be unique as well, because you play differently than others. No matter how you might try to copy them exactly. As I like to repeat (and many have said before me), “Tone is in the fingers”.
Your fingers themselves have a “sound” just like your vocal chords. Part of mastering an instrument is embracing your unique voice and making it more distinctive. Obviously the mic and PA and acoustic properties of the room and the recording equipment and even vocal effects will all have an inescapable effect on the vocal sound heard by the audience. But even if you strip all those away, your voice is uniquely yours. Same goes for any instrument you play.
You’ll sound a lot like your earliest influences, and that’s hard to overcome. Just like it’s hard to eliminate your Southern accent if you learned to speak in the South. But you’ll also sound a lot like your most recent influences too, as that’s what you’re most interested in and currently experienced with. Your style and musical vocabulary will change as your interests and efforts do, but you’ll still have your own distinct, recognizable voice.
And of course voices change over time, for many reasons. Age, illness or injury, or just a large variety of new influences and styles repeatedly being added to the list. But for most people, our voices remain recognizably our own the whole time, despite the changes.
Check out some early blues legends. Big Bill Broonzy, Mississippi John Hurt, Robert Johnson, Reverend Gary Davis, Blind Willie McTell, Blind Blake. They all have extremely unique and distinct guitar voices, despite them all playing flat top, steel string acoustic guitars with no effects.
To expand on this story, I remember reading that Nugent’s theory was part of it was down to physicality - not just how and where EVH strikes the strings, but also how his bones/muscles absorbed the energy. I’m not entirely sure if I entirely buy that or not 
, but I can see how it would affect things.
In a nutshell, I’d say the factors influencing a guitarist’s voice are: imagination, physical playing techniques and equipment used to both make and shape the sounds (amps/cabinets/effects/microphones when recording/guitars, in that order). For example, my sound tends to be fairly aggressive, due my preferring “buzzier” amplifiers, a very specific distortion effect pedal, a very specific guitar pickup type and my personal right-hand style, which involves techniques designed to make the guitar or bass “thump” loudly.
Marc Ribot and Joe Gore have some of the most unique combinations of sound and techniques I’ve ever heard. It’s probably not a coincidence they’ve both spent time in Tom Waits’ band, I suspect.
Indeed, A singer can’t sound like Cher on Believe without using autotune but just because you use autotune doesn’t mean you’re going to sound like Cher.
A good example is Tuck Andress. He basically gutted his guitar. The knobs do nothing, and he installed. . .something. . .in it, but it’s mainly his staccato style that makes him unique.
Yeah, I think a lot of it starts in the guitarist’s anatomy and habits, although effects and amps and whatnot account for most of the flavor.
It occurred to me the other day that a big part of John Entwistle’s sound may have been the result of him playing without keeping his right-hand fingertips close to the strings. I’ve seen him (in videos) reaching out for each fingerstroke, which leads to playing hard and an aggressive and noisy sound. I don’t care for it but will readily acknowledge that it does add somethiing to the hard-rock sound.
You know, it’s one thing to have a signature tone and another to excel at getting different sounds from different equipment. And in that second scenario, as far as I’m concerned, Jimmy Page remains the undisputed, multi-tone, heavyweight champion of the electric guitar!!! 
That’s a good point. When they remastered the Beatles’ albums back in ought-nine, I’d save up, buy one, and listen to it with my ears right between my best speakers. Rinse and repeat.
A friend asked me if re-buying them was worth it, and I called them The “OMG, George Harrison is a genius!” Remasters.
I heard so much that I’d never noticed before. I was amazed at how George could easily shift from one style to another, and change his whole sound as needed.
.
(Oh, and I also thought “Hey, Ringo’s doing a lot more work than I’d thought!” There were fills, and even bongos, that were new to me.)
That’s a timely post for me, as I was thinking about Harrison’s guitar skills just last night. I’ve held The Beatles in very high regard for over 50 years but have only started to appreciate Harrison’s contributions in recent years. Last night, I had to marvel at his backward solo on I’m Only Sleeping. He really nailed it there, and I’ve read it was the first time that the technique had been used in pop music. Another bit of his playing that I like a lot is on Strawberry Fields Forever, under the lyrics toward the beginning of the track. It’s a very, very small detail that only pokes through the mix at a few points, and I love to listen to it. Although both of those examples have more to do with his fingers than his “voice” (technique rather than tone), I agree with you about his versatility insofar as styles and sounds. It sounds to me like he was just as good as Page at getting the best sounds from the right equipment.
There are only a very few guitarists that I think I might be able to recognize by their sound (their “voice” and style of playing) if I heard them on an unfamiliar recording without any context. Post-Beatles George Harrison (after he had really gotten into slide guitar) is one of them. But I don’t think I have a good enough ear to hear what, if anything, is distinctively Harrisonesque in his guitar work with the Beatles, nor to be able to tell just by listening which Beatle played which guitar part on the Beatles’ records.
I can’t find it on YouTube at the moment, but there’s a video that did the rounds of Joe Satriani (a famous instrumental rock “shredder”) playing one of his big songs on a dirt cheap guitar through a practice amp.
It still sounded like him.
Yeah, Lennon and McCartney played well, too, and Harrison was never top-heavy with technique (blistering solos, etc.) or even assertive in a way that might have made his playing more identifiable.
One characteristic of his early playing is that it often sounds like the boom-chuck style I mentioned upthread in relation to Merle Travis and Chet Atkins (although I think Carl Perkins was a bigger influence on Harrison). I say “sounds like” because I think he did it all with a flatpick (the boom-chuck comes from the thumb, with or without a thumbpick). I don’t think Lennon or McCartney ever played in that style, although I could be wrong. The part in Strawberry Fields Forever that I mentioned above is a later example of Harrison playing like that. There’s no boom-chuck, but it’s not really necessary in that context, either, with the bass and rhythm guitar.
Allan Holdsworth is another who had a unique, instantly identifiable sound. He certainly knew about and relied on effects and amps, but that singing quality in his playing came from his hands and his “ultra-legato” playing style.
Upthread, @Chefguy mentioned Tuck Andress, who tells a great story that’s pertinent to this thread. It comes from a lengthy article of his that’s no longer on the Tuck & Patti website, but I’ve found it elsewhere. I see a copyright symbol, so I’ll just post the link and the relevant excerpt.
Here’s the link. Scroll down to the third post.
And here’s the excerpt:
Of course I knew I had a long way to go as a guitarist (I had played less than two years), but I also thought part of the weakness of the sound was my guitar (a Mosrite Ventures model). Its strings were too close together, its action was too low, its frets were too small and flat, its pickups were weak. Along about the third set Tommy Crook came in to see how I was doing, then sat in with his own band on my guitar. It was as if Godzilla had wandered through the club swishing his tail. I realized for the first time that it had nothing at all to do with the guitar. It had everything to do with the guy playing the guitar. He sounded just as overwhelming as when he played his own vintage Gibson archtop. Yet his hands looked just like mine.
Excellent.
If you’re into shredding guitar playing, I recommend trying the “Paul Gilbert Channel” on Pandora. Gilbert is incredible himself (great cover of “It’s All Too Much”), but a lot of examples of instrumental electric guitar with unique voices.
