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This paper presents the implementation of a multiplexed analog readout electronics system that can achieve single-electron counting using Skipper-CCDs with non-destructive readout. The proposed system allows the best performance of the sensors to be maintained, with sub-electron noise-level operation, while maintaining low-bandwidth data transfer, a minimum number of analog-to-digital converters (ADC) and low disk storage requirement with zero added multiplexing time, even for the simultaneous operation of thousands of channels. These features are possible with a combination of analog charge pile-up, sample and hold circuits and analog multiplexing. The implementation also aims to use the minimum number of components in circuits to keep compatibility with high-channel-density experiments using Skipper-CCDs for low-threshold particle detection applications. Performance details and experimental results using a sensor with 16 output stages are presented along with a review of the circuit design considerations.