So the Hilbert space $L^2(R,dx)$ is separable and hence admits a countable (Schauder) basis. However, the Fourier transform rewrites the function as an integral over an uncountable set of functions.
In QM we also thing if position eigenstates $|x\rangle$ as a basis, however, this is also uncountable.
I guess, are Fourier transforms really a new basis, or does the fact that all basis are of the same size break down in infinte vector spaces?
At the intuitive level, it is indeed good to think of this as a change of basis. However, one has to be careful when implementing this intuition in a rigorous treatment (e.g., with Gelfand triples, generalized eigenvectors, the spectral theorem for unbounded operators and whatnot). Definitions like that of (Hamel) bases and theorems about them like, all having the same cardinality, are not to be taken too seriously in this context.
Note that when one chooses a basis in a finite-dimensional vector space $V$, this automatically endows tensor powers $V^{\otimes n}$. Change of bases maps define (in a functorial way) similar maps for tensor powers. This is also true of the Fourier transform and is used a lot in the quantum field theory literature where one often switches from $x$-space representation to momentum space representation for Feynman diagrams and correlation functions. One can also switch to discrete bases, e.g., the basis of Hermite functions. One area where this is particularly useful is group field theory as well as noncommutative field theory (see for example the work of Grosse and Wulkenhaar which reformulates NCQFT in the so-called matrix basis).