Can I run both Windows and Linux Dwarf Fortress from the same directory?

Yes. As far as I can see, and have tested, there should be no problem running both versions from the same (merged) folder. Here is the output of

diff -ur ./df_linux ./df_windows

Only in ./df_windows: compress_bitmaps.bat
Only in ./df_linux/data/art: Thumbs.db
Only in ./df_linux: df
Only in ./df_windows: Dwarf Fortress.exe
Only in ./df_windows: fmod.dll
Only in ./df_windows: fmodex.dll
Only in ./df_linux: g_src
Only in ./df_windows: jpeg.dll
Only in ./df_windows: libpng12-0.dll
Only in ./df_linux: libs
Only in ./df_windows: libtiff-3.dll
Only in ./df_windows: Microsoft.VC80.CRT.MANIFEST
Only in ./df_windows: Microsoft.VC90.CRT.MANIFEST
Only in ./df_windows: msvcm80.dll
Only in ./df_windows: msvcm90.dll
Only in ./df_windows: msvcp80.dll
Only in ./df_windows: msvcp90.dll
Only in ./df_windows: msvcr80.dll
Only in ./df_windows: msvcr90.dll
Only in ./df_windows: msvcrt.dll
Only in ./df_windows: optipng.exe
Only in ./df_windows: optipng license.txt
Only in ./df_linux/raw/graphics/example: Thumbs.db
Only in ./df_linux: README.linux
Only in ./df_windows: SDL.dll
Only in ./df_windows: SDL_image.dll
Only in ./df_windows: zlib1.dll

all other files are identical

(Toady left some Thumbs.db from Windows in the linux version, bad Toady!)

As you can see, the only noticable differences are the libs for the Linux version and DLL files for the Windows version, and that each version has it's own executable, which is key for mutual co-existence (;))

You can even use a modded Windows version of DF and add the required Linux files to it to be able to play it on Linux.