The "2-dimensional" aspect went into the SFT Hamiltonian/potential.
SFT contains the GW invariant (viewing a closed manifold as a cobordism $\varnothing\to\varnothing$, and possibly cut along various contact hypersurfaces). But there is more, because we can study those contact (2n−1)-manifolds and the symplectic 2n-cobordisms between them, and that's what the "n-dimensional field theory" plays with. For GW we analyze the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory.
This algebraic formalism of SFT is clarified in Eliashberg's 2006 ICM talk, "Symplectic field theory and its applications".