I hear a (small) difference between an LP and a similar PRS. To me, that defining tone of the LP isn't so much a low end thing, it's a very sweet and prominent midrange. I actually think a lot of PRS guitars have more low end than a Gibson, it's not that the mids aren't there in a PRS, they just have more bass too, fuller in every register, so the mids don't stand out as much. You can hear that midrange thing in players that have a very straightforward gear philosophy, usually just the "guitar into an amp" guys - Warren Haynes, Derek Trucks, Joe Bonamassa. I think the reason we hear people asking if the PRS will do "that thing" is because they're fed up with the inconsistency of the big G, they want a guitar they can count on with the sound that they want. Or they just want ammunition to wank on about how nothing else can be a Gibson. Until the 594 came along, I didn't often hear a PRS hitting that exact sound, although I think they were certainly close enough that the excessive internet whining was unjustified. But, full disclosure, I have never owned an LP, or a 594. That's just what I hear in clips.
I shopped LP style guitars for several months and had a heck of a time finding something that did what I considered to be a good LP sound - I was on a budget, and of all things, it ended up being an SE model, although I gave Gibson and Epiphone an honest shot. The budget G's and Epiphones just didn't sound right, they were thin and plinky. I did find one Gibson more recently that sounded MASSIVE, but the intonation was insufferable. I'll stick to my SE (and soon my 594).