Thanks for the question. Many other people may have the same question so I thought it was a good one.
The answer is that DMRG treats every system as a 1d chain in the sense that it represents the wavefunction as a matrix product state even for a 2d Hamiltonian. But one can input a 2d Hamiltonian anyway by numbering the sites in a 1d fashion then defining the interactions in the geometry of the 2d system, still using the 1d labeling of sites. From the 1d point of view the Hamiltonian has further-neighbor interactions (up to approximately the size Ny of the system in the y-direction) but DMRG has no fundamental problem dealing with different ranges of interactions. The main issue that occurs is that 2d wavefunctions have fundamentally more entanglement and so the MPS sizes get large.
I've created a code formula for creating a ladder Hamiltonian using AutoMPO: http://itensor.org/docs.cgi?page=formulas/ladder
As you can see there is a 1d ordering of sites (indexed by the variable j) with odd sites corresponding to the first leg of the ladder and even sites corresponding to the second leg.
For a more general idea of how to do "wider" 2d systems you can look at this code formula: