In my opinion, you should attempt to understand the case of incompressible fluids before looking at the compressible case, since it's significantly more involved. You can find a discussion of the latter in "Semidirect Products and Reduction in Mechanics" by Marsden, Ratiu, and Weinstein.
For the incompressible case, the result is originally due to Arnold, so that's a good place to learn it. You could also try Section 1.5 of "Mechanics and Symmetry" by Marsden and Ratiu.
Edit: I should probably add a couple of comments to explain the main idea: the reason the Poisson bracket looks noncanonical is because it is induced by the Kostant-Kirillov-Souriau form on the coadjoint orbits of the volume-preserving diffeomorphism group (this is sometimes called the Lie-Poisson bracket). Although you can locally define canonical coordinates on these orbits, such coordinates aren't often geometrically meaningful (though it is possible to introduce canonically conjugate Clebsch variables for the problem - see here). The KKS symplectic form on coadjoint orbits of a group $G$ can be obtained by applying symplectic reduction to the $T^*G$ with its usual symplectic structure. Hence to see canonically conjugate variables, you need to look at the problem in $T^*Diff_\mathrm{vol}$, where $Diff_\mathrm{vol}$ is the group of volume-preserving diffeomorphisms. This is equivalent to considering the problem in Lagrangian coordinates, as opposed to the reduced picture, which considers the problem in Eulerian coordinates. This is all explained in the book "Mechanics and Symmetry" mentioned above.