Copyright, patents and trademarks are different things. Please don't use them interchangeably.
By registering the Les Paul body design as a trademark, Gibson declared that its shape is as much a part of the Gibson identity as the Gibson name or logo. When someone says "Guitarist [x] plays a Gibson," 9 times out of 10, people will think of the Les Paul. And when someone says "Guitarist [x] plays a Les Paul," you should automatically think "Gibson," but with so many Les Paul copies on the market, that's not automatically the case. This is Gibson's argument, and it's well within their rights to defend their identity that they have built up over decades.
Another key distinction is that copyright and patents both expire after a period of time. Trademarks can remain in effect indefinitely, as long as the trademark is still used in commerce. So trademark registration gives the company a lot more protection.
If i designed a unique instrument shape that became a cultural icon and was exclusively associated with my company, there's no doubt that i would trademark it and vigourously defend it from being ripped off.
What i question is why Gibson didn't go after Tokai or some of the huge Korean manufacturers that are cranking out carbon copies of Les Pauls by the thousands, for anyone who will pay to have their logo stamped on the headstock. Personally, i think the PRS design is different enough that it shouldn't have been seen as an infringement. But clearly Gibson's lawyers were more convincing that PRS's.
Here's another example: Shortly after the iMac made its début, numerous copycat Wintel machines were put out on the market. Apple filed "trade dress" infringement suits against companies like Future Power and e-Machines because their designs were clearly trying to capitalize on the iMac's distinctive design and the image it projected in the marketplace. Apple won.