I was getting ready to post this. The empirical equation for the frequency of the oscillation is on Wikipedia, which should allow easy testing of the hypothesis that a Karman vortex street is responsible if any particular object is thought responsible. As I recall, if the Reynolds number is too high, the oscillation will become broadband, so it won't sound as described.
Of course, other fluid dynamical instabilities could be responsible, so discounting a Karman vortex street doesn't eliminate fluid instabilities in general. A more general fluid dynamical instability is the Kelvin-Helmholtz instability: https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_insta...
Of course, other fluid dynamical instabilities could be responsible, so discounting a Karman vortex street doesn't eliminate fluid instabilities in general. A more general fluid dynamical instability is the Kelvin-Helmholtz instability: https://en.wikipedia.org/wiki/Kelvin%E2%80%93Helmholtz_insta...