Every time this "debate" comes up, people confuse "sound" with "sound." That's not double talk, but it shouldn't be hard to understand that the word has several forms of meaning. The old "Ted Nugent picked up Eddie Van Halen's guitar and plays through his rig, he "sounded" like Ted, not Eddie. It should not be hard to understand, yet there are 10s of thousands of posts debating this on forums all over the world. The "sounds" in that example means "the playing style." Ted doesn't play like Eddie. But, what is coming out of the speakers is 100% Eddie's rig tone. It SOUNDS just like Eddies rig! But the playing style SOUNDS like Ted.
Sorry for the strong opinions on this but the continued debates are things I verified for myself years ago and I shake my head every time I read something like my example above. A fretted note is a fretted note. END OF STORY! Your pick definitely makes a difference, as does pick angle, how hard you pick, etc. So yes, your hands can influence your "SOUND" in that way... but guess what. I can use the same pick, pick angle and pick attack as you if I want too, so once again, I can SOUND just like you in the only way that your hands influence your tone or SOUND.
Now, your phrasing, your vibrato, and the other elements that make up your "style," are what makes you sound like you. But if you do that through my rig, it still SOUNDS like my rig... and you playing.
You could always tell whether it was mom, me, my sister or my dad playing the grand piano even from the other end of the house. But you could also tell when it was mom mom playing the electric piano. Didn't SOUND the same at all, but still SOUNDED like mom.
"So and so sounds like himself no matter what rig he plays through" has to do with STYLE, not tone. Your style is in your hands and fingers (ha, actually, your head!) but TONE is in the rig you play.