As you've noticed, recent versions of OS X only display generic folder icons in the sidebar. So why does Dropbox get special treatment?
The short answer is that Dropbox uses undocumented API to accomplish this. In non-technical speak, it's a special hack that's installed by the Dropbox application.
Some curious folks on StackOverflow found the specific mechanism used by Dropbox: good ol' mach_inject. The same bundle provides both the toolbar item and the sidebar icon.
If you're curious, the resource files live here (at least, on my system):
Clearly, Dropbox goes to great lengths in order to integrate seamlessly into the Finder. Lacking similar treatment, other ordinary folders have only generic icons in the sidebar.