Well, it's a cliche, but a certain amount of tone IS in your hands - where your picking hand falls, whether you fret with just your fingertips or more of the fleshy part of your fingers, little things like that. If everything sounds like a LP to you, I'm guessing you pick a bit closer to the neck than is normal, and tend to err on the side of fretting with the softer, fleshier part of your fingertips? That'd certainly darken your "tone" a little bit...
For kicks once, I recorded a quick backing of the last "Comfortably Numb" solo, mic'd up my amp... ...and then grabbed a new Schecter C7 Hellraiser I'd just picked up that the two guitar forums I post to by and large didn't know I had. Everyone knew I owned an Universe and a Stratocaster, and I posted the clip without much in the way of explanation. Now, a Hellraiser is a 26.5" scale set neck fixed bridge guitar with a set of
EMG 707's, and my strat's a maple neck trem-equipped alder guitar with gold lace sensors, so we're talking night-and-day different. And most of the comments I got were "man, you've got me really gassing for a Stratocaster now." Sure, EMG's are actually voiced a little more like singlecoils than most people would expect, but come on...
But yeah, while I sound like me no matter what I play, if I put down my strat and pick up my Universe, it DOES sound a bit different... Weird that you hear no difference.