"If somebody owns something, it's not mine to take from them, even if I want it really badly."
That's the crux of much of the copyright debate, right there. Copying doesn't take someone's work from them, he or she still has access to it. Right or wrong, good or bad, it's not theft. Unfortunately, much of the public debate surrounding copyright circles around appeals to emotion about how much like theft copyright violation is.
Personally, I'd say not at all. It might be similar conversion -- the use of someone's property without their permission -- but I'm not sure about that.
There's also the "thingness" of a work. In the usual sense, even if your first copy of a work is a thing, it's not the same thing as a copy of it. And copyright necessarily acknowledges that, as the property like component is not the work, but the copyright itself. So the thing that is (sort of) yours -- if it exists enough to call it a thing -- is not the thing you created, it's the legal instrument that controls replication of the thing you created.
You're right, copyright is complicated. But it's not so complicated that confusing it with possession or ownership of physical property is a useful approximation.