It's true that a 2D discrete FFT can also be used to produce an orthogonal expansion of an input matrix, but a truncated reconstruction using it (like a band-pass filtering) won't have the rank properties (e.g. optimal low-rank approximation) of a truncated SVD-based approximation.

They are relatable in many ways if you are interested in data filtering and analysis (viewing matrices as data structures rather than as operators in a function space). Generally the SVD is nicer to have, but the FFT is easier to compute (O(n^3) vs O(n^2 log (n)).

Another way to think about them:

An FFT will describe your data in terms of a fixed orthonormal basis: sine and cosine functions of varying frequency.

An SVD will describe your data using a basis that is also orthonormal, but is entirely based on your data.