The joint communication and sensing (JCS) system can provide higher spectrum efficiency and load saving for 6G machine-type communication (MTC) applications by merging necessary communication and sensing abilities with unified spectrum and transceivers. In order to suppress the mutual interference between the communication and radar-sensing signals to improve the communication reliability and radar-sensing accuracy, we propose a novel code-division orthogonal frequency-division multiplex (CD-OFDM) JCS MTC system, where MTC users can simultaneously and continuously conduct communication and sensing with each other. We propose a novel CD-OFDM JCS signal and corresponding successive-interference-cancelation-based signal processing technique that obtains code-division multiplex gain, which is compatible with the prevalent orthogonal frequency-division multiplex (OFDM) communication system. To model the unified JCS signal transmission and reception process, we propose a novel unified JCS channel model. Finally, the simulation and numerical results are shown to verify the feasibility of the CD-OFDM JCS MTC system and the error propagation performance. We show that the CD-OFDM JCS MTC system can achieve not only more reliable communication but also comparably robust radar sensing compared with the precedent OFDM JCS system, especially in a low signal-to-interference-and-noise ratio regime.