Yes, multidimensional DFT's have complex conjugate symmetry in each dimension
I think you might be misinterpreting the FFTW manual which is referring to internal storage formats.
Edit:
why the transformed vector in FFTW is diminished only by half (only one dimension values are stored on N/2+1 complex numbers)? Why isn't each direction diminished?
From the FFTW manual:
The multi-dimensional transforms of FFTW [...] compute simply the separable product of the given 1d transform along each dimension of the array
So for example if you provide a 256x256 array of real numbers the result will be 256 1 dimensional DFT's. As they mention at the bottom of that documentation this is not a multi-dimensional DFT.
see http://www.fftw.org/fftw3_doc/Complex-Multi_002dDimensional-DFTs.html