Physics of the multiferroic interfaces is currently understood mostly within a phenomenological framework based on screening of the polarization field and depolarizing charges. Additional effects still unexplored are the band dependence of the interfacial charge modulation and the associated changes of the electron-phonon interaction, coupling the charge and lattice degrees of freedom. Here, multiferroic heterostructures of the colossal-magnetoresistance manganite La_1-xSr_xMnO₃ buried under ferroelectric BaTiO₃ and PbZr_xTi_1-xO₃ are investigated using soft-X-ray angle-resolved photoemission. The experimental band dispersions from the buried La1-xSrxMnO3 identify coexisting two-dimensional hole and three-dimensional electron charge carriers. The ferroelectric polarization modulates their charge density, affecting the coupling of the 2D holes and 3D electrons with the lattice which forms large Fröhlich polarons inherently reducing mobility of the charge carriers. Our k-resolved results on the orbital occupancy, band filling and electron-lattice interaction in multiferroic oxide heterostructures modulated by the ferroelectric polarization disclose most fundamental physics of these systems needed for further progress of beyond-CMOS ferro-functional electronics.