UART works normally in loopback mode. However, after connecting MAX13089, half-duplex mode works normally but I am not receiving data in full-duplex mode. What is the cause of this?

The MAX13089 has a full-duplex/half-duplex control pin. When connecting in  full-duplex mode, please confirm that the H/F pin is connected to GND. In addition, this device also has receiver phase/polarity pin. This pin must be correctly set according to the polarity of the input signal in order to ensure correct communication.