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I would think the main engine of D3 shares several components with the SC2 engine. They are both top-down, they both render a similar number of units/monsters on screen, they both zoom in and out to a similar distance. It wouldn't make sense in a time or money perspective to completely write a unique engine for one game or the other.
That said, I'm sure there are MANY different UNIQUE functions, modules, etc tacked on to each game's engine. For instance, SC2 doesn't even have a module for computing random item drops and displaying these drops on screen. Think of games sharing the same - but ultimately different - engines such as Gears of War and Unreal Tournament 3. They both use Unreal Engine 3, but accomplish different end goals.
While they both probably borrow from each other, and more than likely from WoW, I doubt they use the same graphics engine. It's highly likely that they use the same sound, input and networking code though, save for some minor tweaks here and there for each game.
No, they are completely different. SC2 has a lot of bump mapping which I don't see any in Diablo 3. SC2 has an engine designed for roughly around 150 units or so on 1 screen at the same time, and to not drag down as much, especially with the netcode. Diablo 3 won't have nearly as many enemies on screen at a time and they are higher detail than in SC2. The lighting system in both are different. The physics are the same but that's about it. The water effects look different although we've mainly only seen rivers in Diablo 3 but they aren't the same graphics as in SC2. Definitely 2 different engines developed specifically to match both games style of gameplay. Blizzard doesn't go cheap, nor do they lack the resources now.