If an UART is receiving data continuously, can I also send data using the same UART while receiving data at the same time?
Simply, you can.
In addition to charansai's answer, if the MCU supports hardware UART then it becomes easier. Since the transmit/receive process is done via the hardware itself, you don't have to wait for a transmit or a receive process to complete. Depending on the setup, the module raises either an interrupt or a flag once the process is complete.
Even if a software-UART is used, it's still possible by using some programming skills.
Please read basics about UART, at least wiki! Here is the quote from wiki
Communication may be simplex (in one direction only, with no provision for the receiving device to send information back to the transmitting device), full duplex (both devices send and receive at the same time) or half duplex (devices take turns transmitting and receiving).
So, full duplex communication is possible.