Relation of SFT and Gromov-Witten theory
The "2-dimensional" aspect concerns the 2-dimensional curves and their punctures/nodes, and this 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 each n). For GW we're analyzing the marked/nodal points on surfaces and their moduli, and that's the playground of the 2-dimensional field theory (independent of n). So in each case (SFT versus GW) the field theory structure is about different things. The 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, concerning contact 3-manifolds and symplectic 4-cobordisms. It recovers Taubes' Gromov invariant, and there is a forthcoming paper of Hutchings on "ECH as a field theory" (you can find a sketch now in his blog posts). This is expected, because ECH was inspired by SFT and Taubes' Gromov invariant.
What follows is a guess. I don't know enough about Symplectic Field Theory to be sure.
In perturbative string theory, one describes physics in the target spacetime $X$ by summing over maps from Riemann surfaces into $X$. These sums have an interpretation in terms of a 2d CFT living on the Riemann surfaces, but for physicists, the 2d constructions are auxiliary. They're primarily interested in physics in the target space (and indeed, there are constructions of string theory, such as Matrix theory and AdS/CFT, which make no mention of worldsheets).
The physics on the worldsheet is local on the worldsheet, as you point out. The physics on the target spacetime $X$ isn't local in general -- that doesn't seem to be compatible with what we know of gravity -- but it does reduce to local quantum field theory on $X$ in some limits and special situations.
The most obvious example is the low-energy limit, where the target space string theory can be approximated by a gauge theory on $X$ or some compactification of $X$.
N=4 super Yang-Mills theory describes life in the 10d string theory spacetime as perceived by a 3-dimensional D3-brane embedded in the spacetime. (The 6 scalar fields on the 4d worldvolume parametrize the transverse fluctuations of the brane inside the 10d spacetime.)
Perhaps more interesting to mathematicians: Witten's realization of Chern-Simons theory on $M$ as the string theory whose worldsheet CFT is the open-string topological A-model with target space $X = T^*M$ (and appropriate boundary conditions!). This is physically degenerate, but mathematically rather nice: The target space physics is not just local, but actually topological!
You find this sort of construction everywhere in the string theory literature. (Since real string theory calculations are rather hard, most string theory 'consistency checks' involve finding two quantities that can be computed in QFT limits and checking that they match.)
I suspect that symplectic field theory is an example of the third sort: A topologically twisted A-model CFT with appropriate boundary conditions gives rise to a target space string theory which is actually a TQFT, namely symplectic field theory. I'd be more certain of this if I knew exactly which boundary conditions were involved.
The topological twist in the worldsheet CFT also explains the lack of dependence on the target dimension. For untwisted theories, there's a strong constraint on target space dimension. The 'conformal anomalies' of the 2d sigma model and 2d gravity have to sum to zero. In untwisted theories, the anomaly of the 2d sigma model is 3/2 $d$, with $d$ the dimension of the spacetime and the anomaly of 2d gravity is $-15$, leading to $d=10$. In topologically twisted theories, both anomalies vanish, so there's no constraint implied on target space dimension.