There are most certainly specific guns for skeet and trap. I've been hanging around the skeet range the past few months and here is what I can tell you...
Most 'serious' shooters (retired old guys) shoot superposed guns. That means 'over-under' shotguns. They tend to be made by Kolar, Krieghoff, and Perazzi. Cost? About 12 grand. That's right, 12 grand. Next is Browning, specifically the Citori. Then over-unders by SKB. Then it's everything else. Most skeet guns will have 26-inch barrels and be choked with a 'skeet' choke. They will have raised vented ribs and a large focus is put on making sure the gun fits you. Most skeet guns are over/unders. The semi-auto isn't used nearly as often and pumps are almost nonexistent.
As for trap, totally different sport. Barrels are longer and the ribs are higher and the chokes are different. The trap gun is even more expensive. You'd want a stock with adjustable comb, length of pull, and cast.
Can you shoot either with a $275 used 870 Wingmaster? Sure. Will you have fun? At first. But, to be serious and good, at some point, you're going to have to buy specific guns to each sport and they are not cheap.
If nothing else, you MUST have the appropriate chokes for each game. You won't hit crap with your home defense pump that has no choke.