Memristive devices based on the resistive switching mechanism are continuously attracting attention in the framework of neuromorphic computing and next-generation memory devices. Here, we report on a comprehensive analysis of the resistive switching properties of amorphous NbOx grown by anodic oxidation. Besides a detailed chemical, structural and morphological analysis of the involved materials and interfaces, the mechanism of switching in Nb/NbOx/Au resistive switching cells is discussed by investigating the role of metal-metal oxide interfaces in regulating electronic and ionic transport mechanisms. The resistive switching was found to be related to the formation/rupture of conductive nanofilaments in the NbOx layer under the action of an applied electric field, facilitated by the presence of an oxygen scavenger layer at the Nb/NbOx interface. Electrical characterization including device-to-device variability revealed an endurance >103 full-sweep cycles, retention >104 s, and multilevel capabilities. Furthermore, the observation of quantized conductance supports the physical mechanism of switching based on the formation of atomic-scale conductive filaments. Besides providing new insights into the switching properties of NbOx, this work also highlights the perspective of anodic oxidation as a promising method for the realization of resistive switching cells.