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"*.

You also see this in Embedded Contact Homology, between contact 3-manifolds and symplectic 4-manifolds. It can recover Taubes' Gromov invariant! Michael Hutchings will be uploading a paper soon about "ECH as a field theory" (but you can find a sketch in his blog posts). This is satisfying, because ECH was inspired by SFT and Taubes' Gromov invariant (and, unrelated to the point of this question, its relation to Seiberg-Witten theory). There is also a blog post of Hutchings concerning the possibility of extracting ECH from SFT.