How much scale length matters is not something I can say definitively, though I can understand how the nature of the string’s vibrations could be different given that it is under more tension in a Strat than a Les Paul.
I will also say that when I made all my mods to my Jackson years ago, at one point I put the exact same pickup combo in it that I had in my Les Paul Studio at the time. I didn’t post all the pickup changes I made around that time. As for the basic mod, some may recall that I “Les Paulized“ my Jackson, giving me identical controls and electronics as in the LP Studio. I even matched the pot and cap values that I had in the LP Studio.
I was expecting the Jackson to sound very much like the LP Studio, but it didn’t. It was snappier and wasn’t as warm and fat as the LP Studio, even with identical pickups and electronics. Again, my expectation was that they would sound the same…or at least closer than they did. Now, I know the Jackson also has a bolt-on neck, not a set neck, and it has a basswood body and a maple neck. It also has a Floyd Rose and not a stop tail. So, there are a lot of other variables. Can I single out scale length as the only meaningful variable? No…that would require a lot of testing that I do not have the means to accomplish, so I’m withholding too authoritarian of a claim. But, it is a variable that I can point to that does directly affect the string’s vibration as it deals with the tension the string is under.