Simplest analog circuit to compare two song signals?
Cross-correlation is a measure of similarity of two signals. The Fourier transform uses this property to Transform signals versus orthogonal (Sine and Cosine) reference frequencies, expressing the Fourier transform as a linear combination of Sines and Cosines.

Therefore you could simply use a correlation circuit whose inputs are i) the signal of interest; ii) Cosine of varying frequency. Repeat for Sine (ideally in parallel). Then you have measurement of correlation from which you can calculate the Magnitude and Angle as usual.

To see how it works, consider a signal containing the component cos(W1.t) and correlating it with the test signal cos(W1.t) ; i.e. the same frequency. As correlation is simply multiplying and integrating the product will be cos^2(W1.t) = 1/2 + cos(2W1.t). Assuming you test various frequencies across whole integer cycles, this will be the only component that integrates to a non-zero value (i.e. has a DC component). That's how the FT and DFT work.

I've found a very informative article on the link between the DFT and correlation, that really describes it very well.