The magnetotransport behavior inside the nematic phase of bulk FeSe reveals unusual multiband effects that cannot be reconciled with a simple two-band approximation proposed by surface-sensitive spectroscopic probes. In order to understand the role played by the multiband electronic structure and the degree of two-dimensionality, we have investigated the electronic properties of exfoliated flakes of FeSe by reducing their thickness. Based on magnetotransport and Hall resistivity measurements, we assess the mobility spectrum that suggests an unusual asymmetry between the mobilities of the electrons and holes, with the electron carriers becoming localized inside the nematic phase. Quantum oscillations in magnetic fields up to 38 T indicate the presence of a hole-like quasiparticle with a lighter effective mass and a quantum scattering time three times shorter, as compared with bulk FeSe. The observed localization of negative charge carriers by reducing dimensionality can be driven by orbitally dependent correlation effects, enhanced interband spin fluctuations, or a Lifshitz-like transition, which affect mainly the electron bands. The electronic localization leads to a fragile two-dimensional superconductivity in thin flakes of FeSe, in contrast to the two-dimensional high- T c induced with electron doping via dosing or using a suitable interface.