Hmm I always viewed it as sum over orthogonal bases. Sine/Cosine is choice, while there can be many other bases, like Legendre polynomials. Can you explain more how FT is linked to 'a sum over equivalence classes'?
A high level explanation is that a trace counts how many times each equivalence classes appears.
In sense, it is similar to a multiset, it associates a count with some input.
In the case of the Fourier transform, it maps from time domain to frequency domain. In the frequency domain, we can see the amplitude (count) of the signal at each frequency.