Since their discovery, graphene-based systems represent an exceptional playground to explore the emergence of peculiar quantum effects. The present paper focuses on the anomalous appearance of strong infrared phonon resonances in the optical spectroscopy of bilayer graphene and on their pronounced Fano-like asymmetry, both tunable in gated devices. By developing a full microscopic many-body approach for the optical-phonon response we explain how both effects can be quantitatively accounted for by the quantum interference of electronic and phononic excitations. We show that the phonon modes borrow a large dipole intensity from the electronic background, the so-called charged-phonon effect, and at the same time interfere with it, leading to a typical Fano response. Our approach allows one to disentangle the correct selection rules that control the relative importance of the two (symmetric and antisymmetric) relevant phonon modes for different values of the doping and/or of the gap in bilayer graphene. Finally, we discuss the extension of the same theoretical scheme to the Raman spectroscopy, to explain the lack of the same features on the Raman phononic spectra. Besides its remarkable success in explaining the existing experimental data in graphene-based systems, the present theoretical approach offers a general scheme for the microscopic understanding of Fano-like features in a wide variety of other systems.