Everything you're expecting to have layers is flat.
Actually, the windowing system is not flat - internally everything is represented basically as layers and rendered separately. Whatever is behind this web browser window exists separately and has been rendered on it's own by the windowing system. OS X uses a compositor, this rasterizes everything together to be sent off to the framebuffer only at at the very end of the process. Consider that you can take a screenshot of an individual window with the built-in screenshot features of OS X and the alpha-transparent parts of a window and the windows' shadows are preserved in the PNG and magically do not hint of anything behind what was captured.
Considering that it's possible on OS X to capture screenshots of individual windows, it's possible to create layered screenshots.
Tools are able to grab the surface of each window separately and then assemble these together as layers into an exact representation of the entire screen with each layer containing the full window contents and with accurate alphatransparency. Acorn has a feature built-in to do this which is quite nice.
You're able to just delete windows you didn't want to have there, move stuff around, even change out the desktop background behind the desktop icons.
A tool that's free on the Mac App Store is Screenshot PSD, which produces layered PSDs you can open up in Photoshop. There are others.
Regarding extracting layers when you are looking at a "logo" - well, if you took a screenshot with a logo on a website visible with the above utility there would be "layers" in the screenshot but one of them would be the the full web browser window. You could try a utility like Page Layers which would break a websites' elements up into a PSD file with each of them as a layer. But you'd extract a logo file from that just as easily manually by pulling up a web inspector or doing a "save as" on the image.
To get layers that are actually useful for you would require explaining the nature of what you want and what you are dealing with - there are a lot of cool tools but at the end of the day information can't be retrieved if it doesn't exist.