Whether a discretisation is consistent (and the order of consistency) in the FDM setting is defined by the the truncation term, e.g.:
$$\partial_{xx} = \frac{1}{h^2}\begin{bmatrix}1 & -2 & 1\end{bmatrix} + O(h^2)$$
For instance we cannot construct a consistent (wrt the above definition) scheme for $\Delta$ in 2D with a $4$-stencil, and instead at least a $5$-stencil is required. This is not the case for finite element methods, as it is perfectly fine to have $\mathbb{P}_1$ elements where some inner vertex has only 3 neighbours (corresponding to a $4$-stencil, and one can indeed perform the Taylor expansion to verify that the scheme is not consistent wrt the definition for FDM). Instead bounds on the error can be derived per element (e.g. Larson-Bengzon) and in general guaranteeing consistency (for an appropriate mesh).
I am looking for references connecting the two, e.g. applying the FEM analysis to some "inconsistent" FDM stencils, that nevertheless lead to convergence (e.g. are consistent w.r.t. the FEM formulation)?