Recent advances in the development of cryogenic particle detectors such as magnetic microcalorimeters (MMCs) allow the fabrication of sensor arrays with an increasing number of pixels. Since these detectors must be operated at the lowest temperatures, the readout of large detector arrays is still quite challenging. This is especially true for the ECHo experiment, which presently aims to simultaneously run 6,000 two-pixel detectors to investigate the electron neutrino mass. For this reason, we developed a readout system based on a microwave SQUID multiplexer ($μ$MUX) that is operated by a custom software-defined radio (SDR) at room-temperature. The SDR readout electronics consist of three distinct hardware units: a data processing board with a Xilinx ZynqUS+ MPSoC; a converter board that features DACs, ADCs, and a coherent clock distribution network; and a radio frequency front-end board to translate the signals between the baseband and the microwave domains. Here, we describe the characteristics of the full-scale SDR system. First, the generated frequency comb for driving the $μ$MUX was evaluated. Subsequently, by operating the SDR in direct loopback, the crosstalk of the individual channels after frequency demultiplexing was investigated. Finally, the system was used with a 16-channel $μ$MUX to evaluate the linearity of the SDR, and the noise contributed to the overall readout setup.